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CH 301 Unit 3, Exit Ticket 19 Review Notes

1. Chemical Systems & State Functions

Chemical systems can be classified into components known as the universe, which is defined as the combination of the system and its surroundings. Understanding how different properties behave in relation to the system's size and state is crucial:

• Intensive properties:
  These properties are independent of the system size. Examples include:

  • Temperature: A measure of the average kinetic energy of the molecules in a system.

  • Density: The mass per unit volume, invariant regardless of the total mass of the substance.

• Extensive properties:
  These properties depend on the amount of substance present and are additive. Examples include:

  • Pressure: The force exerted by the particles of a substance per unit area.

  • Volume: The amount of space occupied by a substance.

  • Moles: A measure of the quantity of a substance, reflecting the number of particles present.

• State functions:
  These functions depend solely on the initial and final states of the system, not on how the change occurs. Commonly referenced state changes include:

  • $riangle T$ (Change in temperature)

  • $riangle V$ (Change in volume)

  • $riangle P$ (Change in pressure)

  • $riangle n$ (Change in the number of moles)

• Path functions:
  Unlike state functions, path functions depend on the specific process taken to achieve a particular change. Notable path functions include heat and work, which are not dependent solely on the initial and final states but rather on the route taken during the process.

2. Kinetic Molecular Theory (KMT)

The Kinetic Molecular Theory provides a framework for understanding the behavior of gases based on the motions and interactions of their particles:

• Gases consist of tiny, hard-sphere particles:
  These particles are in constant random motion, which accounts for the properties observed in gaseous states.

• Collisions are elastic:
  In elastic collisions, there is no net energy loss; the total kinetic energy before the collision equals the total kinetic energy after the collision.

• Average kinetic energy (Ek) is directly proportional to Temperature (T):
  The relationship is described mathematically as follows:
  $E<em>k = \frac{3}{2} kT$ where $E</em>k$ is the average kinetic energy, and $k$ is Boltzmann's constant. Thus, as temperature increases, the average kinetic energy increases proportionately.

• For gases at the same temperature, lighter molecules move faster:
  This relationship can be articulated by the equation:
  $E_k = \frac{1}{2} mv^2$
  where $m$ is the mass of the gas particles and $v$ represents their velocity. In two different gases at the same temperature, the equation implies that:
  $v \propto \frac{1}{\sqrt{m}}$
  indicating that lighter gas molecules will, on average, travel faster than heavier gas molecules.

3. Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann Distribution describes the distribution of speeds of gas particles in a system:

• Gas particles have a range of speeds due to collisions:
  The motion of gas molecules results in a variety of speeds among the particles, forming a distribution that can be graphed.

• As temperature increases, the speed distribution widens and shifts to higher velocities:
  Higher temperatures result in a greater average speed of particles, causing the distribution to spread out.

• As molecular mass decreases, the distribution spreads wider:
  Lighter molecular masses lead to greater variability in speeds among gas particles, resulting in a broader distribution curve.

• Average kinetic energy increases with temperature:
  This increase in average kinetic energy reinforces the notion that temperature serves as a critical factor in affecting the motion of gas particles.

4. Graham’s Law – Ratio of Gas Speeds, Effusion, & Diffusion

Graham's Law relates the rates of effusion and diffusion of gases to their molar masses:

• From Kinetic Molecular Theory, we know:
  $E_k = \frac{1}{2} mv^2 = \frac{3}{2} kT$. Consequently, can conclude that:
  $\frac{1}{2} mv^2 \propto T$, establishing a relationship between kinetic energy, mass, and velocity.

• For gases at the same temperature:
  This leads to the equation:
  $m<em>1 v</em>1^2 = m<em>2 v</em>2^2$
  which can be rearranged to express the ratio of velocities:
  $\frac{v<em>1}{v</em>2} = \sqrt{\frac{m<em>2}{m</em>1}}$
  indicating that lighter gases will effuse and diffuse faster than heavier gases.

• Effusion and diffusion processes:
  Both processes adhere to the inverse square root relationship established by Graham's Law.

  • Effusion: The process through which a gas escapes from a container through a small hole.

  • Diffusion: The process by which gas molecules spread out in a medium.

• Example:
  Considering oxygen (O2) with a molar mass of 32 g/mol and hydrogen (H2) with a molar mass of 2 g/mol, we find:
  $\frac{v<em>{H2}}{v</em>{O2}} = \sqrt{\frac{32}{2}} = 4$
  Therefore, hydrogen molecules move four times faster than oxygen molecules under the same conditions.