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

For an ideal gas:

1. At constant volume: $(dQ)_V = dU$ (no work done)
2. At constant pressure: $(dQ)_p = dU + p\,dV$
3. For ideal gas: $p\,dV = nR\,dT$

From these equations: $(dQ)_p = (dQ)_V + nR\,dT$

Dividing by $n\,dT$: $\frac{1}{n}\frac{(dQ)_p}{dT} = \frac{1}{n}\frac{(dQ)_V}{dT} + R$

By definition: $C_p = C_V + R$

This shows the molar heat capacity at constant pressure exceeds the molar heat capacity at constant volume by the gas constant R.

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.

Isothermal Process:

• Heat transfer: Heat flows in/out to maintain constant temperature
• Work done: $W = nRT\ln(V_2/V_1)$
• Internal energy change: $\Delta U = 0$ (temperature is constant)
• P-V relationship: $pV = \text{constant}$ (Boyle's Law)
• Molar heat capacity: $C_{isothermal} = \infty$ (heat added produces no temperature change)

Adiabatic Process:

• Heat transfer: No heat exchange with surroundings ($\Delta Q = 0$)
• Work done: $W = \frac{p_1V_1 - p_2V_2}{\gamma-1}$
• Internal energy change: $\Delta U = -W$ (decreases during expansion, increases during compression)
• P-V relationship: $pV^\gamma = \text{constant}$ (steeper curve than isothermal)
• Molar heat capacity: $C_{adiabatic} = 0$ (temperature changes without heat input)

Key differences:

• Adiabatic processes involve temperature changes; isothermal processes don't
• Isothermal processes require heat transfer; adiabatic processes prohibit it
• P-V curves for adiabatic processes are steeper than for isothermal processes
• Work done differs between processes even for the same initial and final volumes

Key physical principles:

• Conservation of energy: Heat lost by gas equals heat gained by water and calorimeter
• Constant pressure conditions: Maintained through precise flow control
• Thermal equilibrium: Gas exits the calorimeter coil at a temperature between initial and final water temperatures

Critical controls and sources of error:

1. Flow rate regulation:

   • Must be kept constant using the screw valve
   • Monitored by maintaining constant levels in the second manometer
   • If flow rate varies, pressure (and thus heat transfer conditions) changes

2. Temperature measurements:

   • Must accurately record bath temperature (θ₁)
   • Initial (θ₂) and final (θ₃) water temperatures determine energy transfer
   • The gas temperature assumption ((θ₂+θ₃)/2) introduces systematic error

3. Heat loss prevention:

   • Calorimeter must be well-insulated
   • Connections must prevent heat exchange with surroundings
   • Gas must have sufficient residence time in coils for thermal equilibrium

4. Gas quantity determination:

   • Pressure drop in the tank must be precisely measured
   • Temperature of gas in tank must remain constant

Joly's differential steam calorimeter determines CV through these steps:

1. Two identical hollow copper spheres are suspended from a sensitive balance and enclosed in a steam chamber
2. One sphere is evacuated while the other contains the test gas
3. When steam enters the chamber, more steam condenses on the gas-filled sphere due to its higher heat capacity
4. The extra mass of condensed water (m₂) is measured by rebalancing
5. CV is calculated using: CV = (M·m₂·L)/(m₁·(θ₂-θ₁)) where:
   • M = molecular weight of gas
   • m₁ = mass of gas sample
   • L = specific latent heat of vaporization
   • θ₁, θ₂ = initial and final temperatures

This method directly measures heat capacity at constant volume because the gas container dimensions remain fixed during heating.

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