Specific Heat Capacities Of Kinetic Theory Of Gases

During adiabatic expansion:

• No heat enters or leaves the system ($ΔQ = 0$)
• Gas does positive work on surroundings as it expands
• From the first law: $ΔU = ΔQ - W = -W$
• Internal energy decreases as work is done
• For ideal gas, internal energy is directly proportional to temperature: $U = nC_V T$
• Decreasing internal energy means decreasing temperature

The gas essentially uses its own thermal energy to perform expansion work, leading to temperature decrease.

This is why compressed gases feel cold when released - they undergo rapid adiabatic expansion.

For an adiabatic process: $dQ = 0$ From first law: $dU + p\,dV = 0$ For ideal gas: $dU = nC_V\,dT$

Therefore: $nC_V\,dT + p\,dV = 0$

Using ideal gas law: $pV = nRT$ Differentiating: $p\,dV + V\,dp = nR\,dT$ So: $dT = \frac{p\,dV + V\,dp}{nR}$

Substituting: $C_V\left(\frac{p\,dV + V\,dp}{R}\right) + p\,dV = 0$ $(C_V + R)p\,dV + C_V V\,dp = 0$ $C_p p\,dV + C_V V\,dp = 0$

Rearranging: $\frac{C_p}{C_V}\frac{dV}{V} + \frac{dp}{p} = 0$ $\gamma\frac{dV}{V} + \frac{dp}{p} = 0$

Integrating: $\gamma\ln V + \ln p = \ln(\text{constant})$ $\ln(p V^\gamma) = \ln(\text{constant})$

Therefore: $pV^\gamma = \text{constant}$

When heat is supplied to a gas at constant volume:

• All heat goes into increasing internal energy (temperature rise)
• No work is done since volume remains fixed

When heat is supplied at constant pressure:

• Gas expands as temperature increases
• Part of the heat is used to do expansion work against external pressure
• Only the remaining heat increases internal energy

Therefore, more heat is required at constant pressure to achieve the same temperature increase as at constant volume, making $C_p > C_V$.

The quantitative difference for ideal gases is exactly the gas constant R: $C_p - C_V = R$

For a reversible adiabatic process:

• $pV^\gamma = \text{constant} = p_1V_1^\gamma = p_2V_2^\gamma$
• Thus $p = \frac{K}{V^\gamma}$ where $K = p_1V_1^\gamma$

Work done: $W = \int_{V_1}^{V_2} p\,dV = \int_{V_1}^{V_2} \frac{K}{V^\gamma}\,dV$

$W = \frac{K}{1-\gamma}\left[\frac{1}{V_2^{\gamma-1}} - \frac{1}{V_1^{\gamma-1}}\right]$

Substituting $K = p_1V_1^\gamma$ and $K = p_2V_2^\gamma$: $W = \frac{1}{1-\gamma}[p_2V_2 - p_1V_1]$

Simplifying: $W = \frac{p_1V_1 - p_2V_2}{\gamma-1}$

This equation allows direct calculation of work using initial and final states without integrating the path.

For a reversible adiabatic process: $TV^{\gamma-1} = \text{constant}$

This means: $T_1V_1^{\gamma-1} = T_2V_2^{\gamma-1}$

Solving for $T_2$: $T_2 = T_1\left(\frac{V_1}{V_2}\right)^{\gamma-1}$

Alternatively, this can be written as: $T_2 = T_1\left(\frac{V_2}{V_1}\right)^{-(\gamma-1)}$

This equation reveals that:

• For expansion ($V_2 > V_1$), $T_2 < T_1$ (cooling)
• For compression ($V_2 < V_1$), $T_2 > T_1$ (heating)
• The temperature change is greater for gases with higher $\gamma$ values

Example: For a diatomic gas ($\gamma = 1.4$) expanding to twice its volume, $T_2 = T_1(0.5)^{0.4} = 0.76T_1$

For an ideal gas, internal energy depends only on temperature, not volume or pressure:

Relationship to temperature:

• $U = nC_V T$
• Where $n$ is number of moles, $C_V$ is molar heat capacity at constant volume, and $T$ is absolute temperature

Determining $C_V$:

• From the relationship: $C_V = \frac{1}{n}\frac{dU}{dT}$
• At constant volume, heat added goes entirely to changing internal energy: $(dQ)_V = dU$
• Therefore: $C_V = \frac{1}{n}\frac{(dQ)_V}{dT}$

For different gas types:

• Monatomic: $U = \frac{3}{2}nRT$ and $C_V = \frac{3}{2}R$
• Diatomic (no vibration): $U = \frac{5}{2}nRT$ and $C_V = \frac{5}{2}R$
• Diatomic (with vibration): $U = \frac{7}{2}nRT$ and $C_V = \frac{7}{2}R$

Note: This direct proportionality between internal energy and temperature is a key property that distinguishes ideal gases from real gases.

Regnault's apparatus measures Cp through these key components and processes:

1. A large pressurized tank containing the experimental gas connects to two copper coils in series:

   • First coil in a hot oil bath (maintained at temperature θ₁)
   • Second coil in a calorimeter with water (initially at temperature θ₂)

2. Working principle:

   • Gas flows through the heated coil, then through the calorimeter coil
   • Gas transfers heat to the water, raising its temperature to θ₃
   • Flow rate is kept constant via a screw valve and monitored with manometers

3. Calculation method:

   • Cp = [(W + m)s(θ₃ - θ₂)]/[n(θ₁ - (θ₂+θ₃)/2)]
   • Where: W = water equivalent of calorimeter, m = mass of water, s = specific heat of water, n = moles of gas passed

4. The amount of gas (n) is determined by measuring the pressure difference in the tank before and after the experiment.

Regnault's apparatus fundamentally cannot measure Cv directly because:

1. Fundamental limitation: The apparatus relies on flowing gas, which requires changing volume and pressure, making constant volume conditions impossible.

2. Required modifications for Cv measurement:

   • Replace flow-based system with sealed chambers
   • Implement a differential calorimeter like Joly's device with two spheres
   • Add precise mechanisms to measure heat without volume change

3. Alternative approach:

   • First measure Cp using the existing setup
   • Measure the gas constant R independently
   • Calculate Cv using the relationship: Cp - Cv = R (for ideal gases)
   • Or determine γ = Cp/Cv through sound velocity measurements

4. Theoretical relationship:

   • For ideal gases: Cv = (1/n)(dU/dT), where internal energy changes without volume change
   • Cp = Cv + R because at constant pressure, additional energy goes into expansion work
   • Different experimental approaches are needed when volume must remain constant

Advantages of differential calorimeter method:

• Direct measurement of CV without conversion from CP
• Compensates for container heating by using identical spheres
• High precision due to differential measurement technique
• Works well for gases that are difficult to handle in flow systems

Sources of error:

• Slight expansion of metal spheres when heated (changes volume)
• Incomplete evacuation of reference sphere
• Uneven steam distribution around both spheres
• Water droplets forming on suspension wires (mitigated by heating coils)
• Imperfect thermal isolation from surroundings
• Non-ideal gas behavior at higher pressures

The method is particularly valuable for measuring CV of gases where P-V work would complicate other measurement approaches.

To determine γ = CP/CV from differential steam calorimeter data:

1. Calculate CV directly from the experiment using: CV = (M·m₂·L)/(m₁·(θ₂-θ₁)) where:

   • m₂ = extra mass of condensed steam
   • m₁ = mass of gas sample
   • M = molecular weight of gas
   • L = latent heat of vaporization
   • θ₁, θ₂ = initial and final temperatures

2. Calculate CP using the relation: CP - CV = R (gas constant)

   • CP = CV + R

3. Compute γ = CP/CV

4. For diatomic gases, expect γ ≈ 1.4 if only translational and rotational degrees of freedom are active

   • If vibrations are active, γ will be closer to 1.29
   • Deviation from expected values indicates additional energy storage modes

5. Use the result to verify the equipartition theorem prediction for the gas's molecular structure