Heat Capacity and Specific Heat: Constant Volume and Constant Pressure

Heat Capacity: Constant Volume vs. Constant Pressure

• It is crucial to specify whether "heat capacity" refers to constant volume or constant pressure, as these are distinct quantities that yield different numerical values.

• We will define two primary heat capacities: one at constant volume ($C<em>v$) and another at constant pressure ($C</em>p$).

Heat Capacity at Constant Volume ($C_v$) for Monatomic Gas

• In a constant volume process, the change in volume ($dV$) is zero, which implies that no work ($W$) is done.

• According to the First Law of Thermodynamics, $dE = dQ - dW$. If $dW = P dV = 0$, then $dE = dQ$. This means that any added heat directly contributes to the change in the system's internal energy.

• Heat capacity at constant volume is formally defined as $C<em>v = \left(\frac{dQ}{dT}\right)</em>V$.

• Given $dQ = dE$ at constant volume, $C<em>v$ can also be expressed as $C</em>v = \left(\frac{dE}{dT}\right)_V$.

• The internal energy ($E$) for an ideal monatomic gas is given by $E = \frac{3}{2}NkT$ (where $N$ is the number of atoms and $k$ is Boltzmann's constant) or, more conveniently for macroscopic systems, $E = \frac{3}{2}nRT$ (where $n$ is the number of moles and $R$ is the ideal gas constant).

• To derive $C<em>v$, we take the derivative of the internal energy with respect to temperature ($T$) at constant volume: $C</em>v = \frac{d}{dT}\left(\frac{3}{2}nRT\right) = \frac{3}{2}nR$

• Thus, for a monatomic gas, the heat capacity at constant volume is $C_v = \frac{3}{2}nR$.

• This quantity represents the total heat required to change the temperature of the entire system. It depends linearly on the number of moles ($n$); a larger number of moles implies a higher heat capacity and, consequently, more energy needed to achieve a given temperature change (e.g., heating a small container vs. a whole room).

Molar Specific Heat at Constant Volume ($c_{v,m}$)

• Molar specific heat at constant volume ($c_{v,m}$) is defined as the amount of heat needed to raise the temperature of one mole of a substance by one Kelvin (or Celsius) degree at constant volume.

• It is related to the total heat capacity at constant volume by $c<em>{v,m} = \frac{C</em>v}{n}$, or conversely, $C<em>v = n c</em>{v,m}$.

• For a monatomic gas, substituting the derived $C<em>v$: $c</em>{v,m} = \frac{\frac{3}{2}nR}{n} = \frac{3}{2}R$

• Using the ideal gas constant $R = 8.31 \text{ J/(mol K)}$, the molar specific heat for a monatomic gas is calculated as:
  $c_{v,m} = \frac{3}{2} \times 8.31 \text{ J/(mol K)} = 12.465 \text{ J/(mol K)}$ (approximately $12.45 \text{ J/(mol K)}$ as presented in the lecture).

Molar Specific Heat at Constant Volume ($c_{v,m}$) for Diatomic Gas

• At typical temperatures, a diatomic gas possesses $5$ degrees of freedom (3 translational and 2 rotational).

• Its internal energy is given by $E = \frac{5}{2}nRT$.

• Following the same derivation logic as for a monatomic gas, the heat capacity is $C_v = \frac{5}{2}nR$.

• Therefore, the molar specific heat for a diatomic gas is:
  $c_{v,m} = \frac{\frac{5}{2}nR}{n} = \frac{5}{2}R$

• Numerically: $c_{v,m} = \frac{5}{2} \times 8.31 \text{ J/(mol K)} = 20.775 \text{ J/(mol K)}$ (approximately $20.75 \text{ J/(mol K)}$ as presented).

• It is more practical and easier to remember these expressions (e.g., $\frac{3}{2}R$, $\frac{5}{2}R$) than the specific numerical values, as the numerical value can be easily calculated by plugging in $R$. These values are essential for calculating heat required for temperature changes.

General Formula for Molar Specific Heat at Constant Volume ($c_{v,m}$) based on Degrees of Freedom (DOF)

• The molar specific heat at constant volume can be generalized to any ideal gas based on its number of active degrees of freedom ($DOF$):
  $c_{v,m} = \frac{DOF}{2}R$

• Examples of DOF:

  • Monatomic gas: $DOF = 3$

  • Diatomic gas: $DOF = 5$

  • Polyatomic gas: $DOF = 6$ or more, depending on molecular geometry and activated vibrational modes.

Experimental Validation for Gases

• The theoretical values for molar specific heat show excellent agreement with experimental measurements.

• Monatomic Gases:

  • Argon: Experimentally measured $12.4717 \text{ J/(mol K)}$, which is very close to the theoretical value of $\frac{3}{2}R \approx 12.45 \text{ J/(mol K)}$.

  • Helium: Measured $12.4717 \text{ J/(mol K)}$, confirming the $\frac{3}{2}R$ prediction.

  • Neon: Measured $12.4717 \text{ J/(mol K)}$, also in agreement with $\frac{3}{2}R$.

• Diatomic Gas (using Air as a proxy):

  • Air (composed mainly of diatomic oxygen and nitrogen): Measured $20.85 \text{ J/(mol K)}$, which is very close to the theoretical $\frac{5}{2}R \approx 20.75 \text{ J/(mol K)}$ for diatomic gases.

• Polyatomic Gas (Carbon Dioxide - $CO_2$):

  • $CO_2$ is a linear molecule, typically exhibiting 3 translational, 2 rotational, and potentially 2 vibrational degrees of freedom, giving it a total of $7$ DOF. (It does not pick up a third rotational degree of freedom because its linearity means rotation around its axis doesn't change its configuration).

  • Measured $28.46 \text{ J/(mol K)}$, which is very close to $\frac{7}{2}R \approx 29.085 \text{ J/(mol K)}$.

• This consistent agreement between theory and experiment validates the model used to derive these specific heat values.

Molar Specific Heat at Constant Volume ($c_{v,m}$) for Solids

• Unlike gases, atoms in solids are locked in a crystalline structure and do not exhibit significant translational or rotational motion.

• Their primary motion is vibration. Each atom can vibrate in three dimensions, and each vibrational mode (oscillator) contributes two degrees of freedom (one for kinetic energy and one for potential energy).

• Thus, per atom, there are $3 \text{ dimensions} \times 2 \frac{DOF}{\text{dimension}} = 6$ degrees of freedom for vibrations.

• For solids, applying the general formula, $c_{v,m} = \frac{6}{2}R = 3R$.

• Numerically: $c_{v,m} = 3 \times 8.31 \text{ J/(mol K)} = 24.93 \text{ J/(mol K)}$ (approximately $24.9 \text{ J/(mol K)}$).

• Experimental Validation for Solids (Metals):

  • Iron: Measured $25.09 \text{ J/(mol K)}$, which is close to $3R$.

  • Copper: Measured $24.47 \text{ J/(mol K)}$, also close to $3R$.

  • Gold: Measured $25.42 \text{ J/(mol K)}$, confirming the $3R$ approximation.

• Exceptions and Limitations: For more complex materials or at extreme temperatures, this simple $3R$ rule may break down. For example, graphite measures $8.53 \text{ J/(mol K)}$ and diamond measures $6.115 \text{ J/(mol K)}$, which are significantly different from $3R$. In such cases, specific values must be looked up in tables.

• These values are generally valid at