During an adiabatic expansion the temperature of 0.600 mol of oxygen drops from 30 degree celsius to 10.0 degree celsius.

Question

1: how much work does the gas do
2: how much heat is added to the gas

Answer 1

To analyze the adiabatic expansion of the gas and answer your questions, we can use the ideal gas law and the principles of thermodynamics.

1. Work Done during Adiabatic Expansion

For an adiabatic process for an ideal gas, we can express the relationship between pressure, volume, and temperature using the following equations:

\[ \frac{P_1 V_1^\gamma}{T_1^{(\gamma-1)}} = \frac{P_2 V_2^\gamma}{T_2^{(\gamma-1)}} \]

Where:

• \( \gamma \) (gamma) is the heat capacity ratio (\( C_p/C_v \)), which is approximately 1.4 for diatomic gases like oxygen (O₂).
• \( P \) is the pressure,
• \( V \) is the volume,
• \( T \) is the temperature in Kelvin.

In an adiabatic process, we can derive an expression for the work done \( W \) by the gas during the expansion. However, we do not have enough information (like the initial and final pressures or volumes) to calculate the work done directly without the pressure and volume changes.

We can use the relation for work done in an adiabatic process for an ideal gas given by:

\[ W = \Delta U = n C_v (T_f - T_i) \]

To compute the work done:

1. Convert the temperatures from Celsius to Kelvin:

   • \( T_i = 30^\circ C + 273.15 = 303.15 , \text{K} \)
   • \( T_f = 10^\circ C + 273.15 = 283.15 , \text{K} \)

2. Calculate \( \Delta U \):

   • For a diatomic gas (like O₂), the molar heat capacity at constant volume \( C_v \) is approximately \( \frac{5}{2} R \) where \( R \) is the ideal gas constant (8.314 J/(mol·K)).
   • \( C_v \approx \frac{5}{2} \times 8.314 \approx 20.785 , \text{J/(mol·K)} \)

3. Now we can calculate the change in internal energy: \[ \Delta U = n C_v (T_f - T_i) = 0.600 , \text{mol} \times 20.785 , \text{J/(mol·K)} \times (283.15 - 303.15) , \text{K} \] \[ \Delta U = 0.600 \times 20.785 \times (-20) \approx -252.36 , \text{J} \]

Since, by the first law of thermodynamics, because we have an adiabatic process and no heat is added or removed, the work done by the gas is equal (but opposite in sign) to the change in internal energy: \[ W = -\Delta U \approx 252.36 , \text{J} \]

2. Heat addition to the gas during the process

In an adiabatic process, there is no heat exchange with the surroundings, meaning that:

\[ Q = 0 , \text{J} \]

Summary

1. Work done by the gas during adiabatic expansion: 252.36 J
2. Heat added to the gas: 0 J

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