Chapter 2: The First Law of Thermodynamics

- energy - work - heat - 1st law - enthalpy - internal energy - heat capacities - changes in state functions for isotherm., ad., irr., & rev. processes

first law of thermodynamics

• there is an extensive state function E (total energy of sys.) for any process in closed sys.

• ΔE is energy change undergone by sys. in process

• change in energy of sys. ΔE happens along change in energy of surroundings −ΔE

• total energy of a system + surroundings remains constant (is conserved)

• ΔU = Q + W (restrict to closed sys., at rest, no ext. fields)

• E = U

what do the heat capacities at constant volume and pressure tell us?

they give the rates of change of the internal energy and enthalpy with temperature

what is 1 N

1 N = 1 kg*m/s²

what is the unit for work?

joules, J

what is a joule? 1J=?

1J = 1N*m = 1 kg*m²/s²

if F is the total force acting on a particle from point 1 to 2, how can u find the total work done?

the total work done will be from the integration of F*dx from point 1 to 2 in each direction summed up

how to calculate work using kinetic energy?

work is the change in kinetic energy on a system

w = K₂ — K₁

what does the work-energy theorem say?

the work done on a particle by a force equals the change in kinetic energy of the particle

dw is the infinitesimal work, to get the total amount of work, what do u do?

an infinite amount of infinitesimal are summed up to calculate integral

so integration is done

closed system + reversible process

what is the infinitesimal change in work?

what is the total work done on system?

dw_rev = −P*dV

w_rev = −∫(1 to 2)P*dV

for work in a closed system + reversible process, what is the sign of work for expansion and compression?

expansion: work is negative, -ve, w<0

contraction: work is positive, +ve, w>0

what is a reversible process?

• idealization

• process where system is always infinitesimally close to equilibrium

• infinitesimal change in conditions can reverse process to restore system + surroundings to initial states

what is the work done due to a volume change called?

P-V work

the value of the line integral is the area under..?

the curve that plots P vs. V

what does the work depend on?

work depends on process used to go from state 1 to state 2

what will happen when 2 bodies at unequal temps. are placed in contact?

they will eventually reach thermal equilibrium at a common intermediate temp

• conventionally, heat has flowed from hotter to colder body

specific heat capacities of substances are functions of what?

temperature and pressure, T and P

What does this equation: dq_P = m*c_P*dT; mean in words

in a closed system, when infinitesimal amount of heat dq_P flows at constant pressure, P into a body of mass m and specific heat capacity at constant pressure c_P the body’s temp is raised by dT

when is heat flow reversible or irreversible?

• heat flow is reversible when the temperature difference btwn the bodies is infinitesimal

• heat flow is irr. when there’s finite temp. difference btwn bodies

what is the internal energy, U?

internal energy is energy that exists at molecular level and includes molecular kinetic and potential energies

internal energy consists of molecular translation, rotational, vibrational, and electronic energies

molar internal energy symbol, calculation, units

• U_m = U/n

• n = number of moles

• J/mol

when is q negative or positive?

• when heat flows into sys. from surroundings during a process, q>0

• when outflow of heat from sys. to surrounds, q<0

what is the internal energy for a closed system for an infinitesimal process?

dU = dq + dw

U is a state function, what does this mean?

the change in U only depends on the final and initial states of the sys., its independent of path used to bring sys. from initial to final state

what is the process called when the final state of the system is the same as the initial state?

cyclic process, ΔU = 0

what are heat and work defined in terms of?

• heat and work are NOT forms of energy

• heat and work are forms of ENERGY TRANSFER

• heat and work exist only in terms of processes

• they don’t exist before or after process of energy transfer btwn sys. and surro.

• heat is energy transfer btwn sys. and surro. due to temp difference

• work is energy transfer btwn sys. and surro. due to macroscopic force acting thru distance

heat is work done at the _________ _______

molecular level

what is “heat flow” in actual thermodynamic terms?

“heat flow“ is energy flow due to temp diff.

units of heat

joule, J

for heat, 1 cal = ?J

1 cal = 4.184J

what is the equation for enthalpy

H = U + PV

units of enthalpy

joules, J

for P-V work, in a closed system, with constant pressure

what is the equation for a change in enthalpy? for what kind of changing work?

ΔH = q_p

only for volume-change work

what is the enthalpy change for a change of state?

ΔH = U₂ + P₂V₂ − (U₁ + P₁V₁)

ΔH = ΔU + Δ(PV)

what is the enthalpy change for a constant pressure process?

P₂ = P₁ = P

Δ(PV) = PV₂ − PV₁

ΔH = ΔU + PΔV

Δ(PV) is different from … and …

P*ΔV and ΔP*V

because only the change in one term is taken

as opposed to the difference in PV altogether

molar enthalpy symbol, calculation, units

• H_m = H/n = (U + PV)/n = U_m + PV_m

• enthalpy is molar internal energy + pressure times molar volume

what is the first law for a constant-volume process

closed sys., P-V work only, V constant

ΔU = q_v

q_v is where heat absorbed at constant volume

what is the heat capacity of a closed system for an infinitesimal process?

C_pr = dq_pr/dT

what does: C_pr = dq_pr/dT mean in verbal terms?

• it means the heat capacity is the heat flowing into the sys. and the temp. change of sys. in the process

• the subscript means that heat capacity depends on the nature of the process

what is the formula for isobaric heat capacity of a closed system for a constant-pressure process?

C_P = dq_P/dT

the subscript indicates that pressure is constant

what is the formula for isochoric heat capacity of a closed system for a constant-pressure process?

C_V = dq_V/dT

the subscript indicates that volume is constant

what does: C_V = dq_V/dT mean in verbal terms?

• it means the heat capacity of a closed system during an infinitesimal constant-volume process.

• it refers to the heat flowing in the sys. and the temp change of the sys. during process

what kind of process do formulas for heat capacity of closed system work for?

C_V = dq_V/dT

C_pr = dq_pr/dT

they work for reversible processes

for an infinitesimal process:

dq_P = dH at const. P

dq_V = dU at const. V

so what can the formula for heat capacities be written as for a closed system in equilibrium, for P-V work?

C_V = (dU/dT)_V

C_P = (dH/dT)_P

what do these equations directly tell us (rate of change) ?

C_V = (dU/dT)_V

C_P = (dH/dT)_P

they give us the rates of change of H and U with temperature

any state function has a definite value once what is specified?

one the system’s state is specified

for a closed system of fixed composition, the state is specified by which functions/variables?

the state is specified by the properties and functions of P and T

any state function of a closed equilibrium system of fixed composition is a function of what?

T and P

what is C_P a function of?

C_P is a function of T and P

what is H (enthalpy) a function of?

H is a function of T and P

what is U (internal energy) a function of?

T and V

what is C_V a function of?

C_V is a function of of T and V

in “specific heat capacity”, “specific volume” and “specific enthalpy”, what does specific refer to? and how to obtain these values?

• specific means “divided by mass”

• c_P, v, and h are obtained by dividing C_P, V, and H by m

which of these are intensive or extensive properties?

• c_P

• C_P

• C_P,m

• c_P = specific heat capacity, intensive

• C_P = heat capacity, extensive

• C_P,m = molar heat capacity, intensive

for a closed system C_P and C_V must be positive/negative?

both must be positive

C_P,m = C_P/n

what is the formula for? and for what kind of sys.?

molar heat capacity for a pure substance

c_P = C_P/m

what is the formula for? and for what kind of sys.?

specific heat capacity for one-phase system

(dU/dV)_T

what type of function is it?

what is it sometimes called?

how does it relate internal energy and volume?

• it is a state function

• sometimes called the internal pressure

• relates to part of internal energy that occurs bc of intermolecular potential energy

• change in system’s volume will change average intermolecular distance and thus average intermolecular potential energy

how do gases, liquids, solids and pressure relate to (dU/dV)_T and whether it is big or small?

• for gas, at low pressure, intermolecular forces are smaller

• this makes (dU/dV)_T smaller

• for liquids and solids, pressure is higher, molecules are closer to e/o

• intermolecular forces are larger

• this makes (dU/dV)_T larger

what is the sign of work when:

• surroundings do work on system

• system does work on surroundings

• w>0, positive

• w<0, negative

what is enthalpy as a concept, in words?

enthalpy is a measurement of energy in a thermodynamic system

enthalpy is the sum of internal energy and the product of pressure and volume of sys.

The Joule and Joule–Thomson experiments measure (dT/dV)_U and (dT/dP)_H; what are these derivatives closely related to?

closely related to (dU/dV)_T and (dH/dP)_T

switch the top(numerator) of derivative with subscript to find close relations

what does a perfect/ideal gas obey?

• PV = nRT

• (dU/dV)_T = 0 (internal pressure)

• (dH/dP)_T = 0

• U, H, C_v, C_P depend only on T

• C_P - C_V = nR

what did the joule experiment demonstrate for real + ideal gas?

the internal energy only depends on temperature T, not volume V

what processes are the changes in thermodynamic properties for ideal gas calculated for easily?

• reversible isothermal

• reversible adiabatic

line integral _L∫(1 to 2)db is independent of path from state 1 to state 2 if …?

if b is a state function

if ∮db = 0 for every cyclic process, then b is a state function, the line integral, ∮db = 0 for what type of process?

• cyclic process

the heat capacities are measures of how much _______ must be added to substance to give a certain _________ in __________

energy must be added to substance to give a certain increase in temperature

the more ways (translation, rotation, intermolecular interactions) a substance has of absorbing added energy, what will happen to its molar heat capacity values?

the greater its heat capacities with constant pressure and volume will be

what happens during the joule experiment? how is work calculated? what is the joule coefficient?

• it’s abt measuring the T change after free expansion of gas into a vacuum

• first chamber A has gas, during equilibrium, the temp is measured by thermometer

• system has adiabatic walls, q = 0, no heat flows in/out

• expansion into vacuum is irreversible

• finite unbalanced forces act within system + lack of pressure equilibrium

• dw = -PdV doesn’t apply, only motion is within system

• gas do no work on surro., surro. do no work on gas

• w = 0 for free expansion into vacuum

• calculate work for closed system, ΔU = q + w

• ΔU = q + w = 0 + 0, constant-energy process

• experiment measure T change w change in V at constant U

• joule coefficient: μ_J = (dT/dV)_U

• experiment gave μ_J = 0 = (dT/dV)_U, improved version was done = JT experiment

how is μ_J = (dT/dV)_U is related to (dU/dV)_T ?

• the variables in partial derivatives r same; T, U, V

• use property (dT/dU)_V * (dU/dV)_T * (dV/dT)_U = −1

• (dU/dV)_T = −C_V*μ_J

what was the joule-thompson experiment? how was work calculated?

• version of joule experiment to get more accurate results

• involved slow pushing of gas through rigid, somewhat permeable plug

• system was closed in adiabatic walls

• left piston had fixed P = P₁, right piston had fixed P = P₂ < P₁

• want to calculate work done on gas by throttling through plug

• process is irr. bc P1 > P2 by finite amount and infinitesimal change in P can’t reverse process

  • but P drop occurs almost only in plus

  • plus is rigid, gas does no work on plug + vice versa

• exchange of work btwn sys. and surro. happens mostly at 2 pistons

• P eq. is maintained at piston, can use dw = −PdV to. get work at each piston

• initial + final volume of left chamber: V₁, 0

• right chamber: 0, V₂

• w_left = −∫(V₁ to 0)P₁dV = −P₁∫(V₁ to 0)dV = −P₁(0 − V₁) = P₁V₁

• w_right = −∫(0 to V₂)P₂dV = −P₂∫(0 to V₂)dV = −P₁(V₂ − 0) = −P₂V₂

  • gas in right chamber does work on piston = negative work

• total work done, w = w_left + w_right = P₁V₁ − P₂V₂

for gases, (dU/dV)_T is nonzero but its _____

small

how was the change in enthalpy calculated for a JT experiment?

• first law for adiabatic process, q=0, due to walls

• ΔU = q + w = w, ΔU = P₁V₁ − P₂V₂ , U₂ − U₁= P₁V₁ − P₂V₂

• rearrange to get U₂ + P₂V₂= U₁ + P₁V₁

• H = U + PV, so H₂ = H₁, ΔH = 0

how is the joule-thompson coefficient defined?

• μ_JT = (dT/dP)_H

• μ_JT is ratio of infinitesimal changes in 2 intensive properties, therefore, it’s an intensive property anf function of T and P

• measurement of T change in JT experiment gives ΔT/ΔP at constant H

what are the conditions for a gas to be cooled by a JT expansion? what is sign of the pressure change over JT expansion

• the conditions are that the JT coefficient must be positive over range of T and P involved

• pressure change is less than 0, pressure would be decreasing

• ΔP < 0, μ_J > 0

in a J-T expansion, the initial and final enthalpies are?

the same/equal

what are values for (dT/dP)_H found for J-T expansion?

• start with initial P₁ and T₁

• pick P₂ less than P₁

• plot points (T₁, P₁) and (T₁, P₂)

• repeat with same initial point (T₁, P₁), and different final pressures

• leads to multiple points that are for states of equal enthalpy

• join points to make curves

• slope of curve at any point gives (dT/dP)_H for the T and P at that point

what is an ideal gas?

an ideal gas has no intermolecular forces

the molecules occupy negligible space

why do we expect that for an ideal gas, U will not change with V at constant T and (dU/dV)_T = 0?

• if change V while holding T constant, average distance btwn mlcls changes but intermolecular forces are 0, so this will not affect U

• average translational kinetic energy of gas mlcls is function of T only, will not change with volume, V

for closed sys. in equilibrium, internal energy (+ any state function) is expressed as function of T and V. but for perfect/ideal gas, U is independent of ____, so U = ?

• U is independent of V

• U = U(T)

for perfect/ideal gas, U is independent of V, so C_V = (dU/dT)_V becomes:

C_V = ?, dU = ?

C_V = dU/dT

dU = C_V*dT

C_V of a perfect/ideal gas depends only on T, so C_V = ?

C_V = C_V(T)

using eq. for ideal gases, for perfect gas, H = U + PV = ?

H = U + nRT

for an ideal gas, enthalpy only depends on ?, H = ?

enthalpy only depends on T, H = H(T)

for perfect/ideal gas, H is independent of P, so C_P = (dH/dT)_P becomes:

C_P = ?

C_P = dH/dT

C_P = C_P(T)

for a perfect/ideal gas, PV = nRT, so (dV/dT)_P = ?

(dV/dT)_P = nR/P

for an ideal gas:

(dV/dT)_P = nR/P

C_P - C_V = P((dV/dT)_P

so C_P - C_V = ?, C_P,m - C_V,m = ?

C_P - C_V = nR

C_P,m - C_V,m = R

how to calculate work for a reversible isothermal process in a perfect/ideal gas, fixed amount of perfect gas?

• isothermal = constant T process

• perfect/ideal gas: PV = nRT

• U = U(T)

• ΔU = 0 for isothermal change of state, ΔT = 0

• ΔU = q + w, 0 = q + w

• q = −w, w = −q

• dw_rev = −PdV, dw_rev = −(nRT/V)*dV

• w = −∫(1 to 2)(nRT/V)*dV

  • = −nRT*∫(1 to 2)(1/V)*dV

  • if there was no 1/V, it would just be (V₂ − V₁)

  • since 1/V is there, use ln

• w = −nRT(lnV₂ − lnV₁)

• w = −nRT*ln(V₁/V₂) = −nRT*ln(P₂/P₁)

• cuz PV = nRT, P and V are proportional, but opposite

for a reversible isothermal process in a perfect/ideal gas, fixed amount of perfect gas

if the process is an expansion, what is the sign of w, q? and what does it rep.?

• expansion, V₂ > V₁

• w<0, negative, work done on system

• q>0, positive, heat added to gas

describe in words what will happen for reversible isothermal process in perfect gas. using visual of gas in cylinder with const. T bath, with frictionless piston.

how is the external pressure changed?

what happens when pressure is increased or decreased?

• the external pressure is changed at an infinitesimal rate

• when the pressure is increased

  • gas is compressed

  • work is done on it, energy transfer to gas

  • temp. increases at infinitesimal rate

  • excess energy leaves as heat flows out of gas to const. T bath

  • maintains const. T for gas

• when the pressure is decreased

  • the gas expands

  • gas does work on surro., energy transfer to surro.

  • temp. decreases

  • causes heat to flow into gas from bath

  • maintains const. T in gas

how to calculate change in internal energy for a reversible adiabatic process in a perfect/ideal gas?

• adiabatic, dq = 0, q = 0

• reversible process in sys. w/ only P-V work, dw = = −PdV

• dU = C_VdT, dU = dq + dw, ΔU = q + w = w

• C_VdT = −PdV = −(nRT/V)dV

• C_V,mdT = −(RT/V)dV

• (C_V,m/T)dT = −(R/V)dV = −∫(1 to 2)(R/V)*dV = R ln(V₁/V₂)

• adiabatic, dq = 0

• dU = C_VdT

• dU = dq + dw

• ΔU = q + w = w

• ΔU = C_V(T₂ − T₁)

what is dU for an ideal gas for any process (no matter const. P, const. V, or not)?

dU = C_VdT

what is dH for an ideal gas for any process (no matter const. P, const. V, or not)

dH = C_PdT

for an ideal gas, C_V,m is a function of T, if T change is small, C_V,m can be taken as..?

approx constant

C_V,m is nearly constant for what type of gas?

monatomic gases

for a perfect/ideal gas, adiabatic process, with C_V constant, what is the relation ship between T₁, V₁, T₂, V₂, R, C_V,m ?

(T₂/T₁) = (V₁/V₂)^ R/C_V,m

what is the heat capacity ratio gamma γ equation?

γ = C_P/C_V

if (contour integral) ∮db = 0 for every cyclic process, what is b?

b is a state function