To analyze the behavior of an ideal monoatomic gas as it expands to twice its initial volume under the conditions given, we need to apply the principles of thermodynamics and the ideal gas law. Let's break this down step by step for both scenarios: when \( PV^2 = \text{constant} \) and \( PV^{4/3} = \text{constant} \).
Understanding the Ideal Gas Law
The ideal gas law is expressed as:
PV = nRT
Where:
- P = pressure of the gas
- V = volume of the gas
- n = number of moles of the gas
- R = ideal gas constant
- T = temperature of the gas in Kelvin
Case 1: Expansion with \( PV^2 = \text{constant} \)
In this scenario, we have:
P_1 V_1^2 = P_2 V_2^2
Given that the volume doubles, we can express this as:
V_2 = 2V_1
Substituting this into the equation gives:
P_1 V_1^2 = P_2 (2V_1)^2
This simplifies to:
P_1 V_1^2 = P_2 \cdot 4V_1^2
From this, we can derive:
P_2 = \frac{P_1}{4}
Temperature Change
Using the ideal gas law, we can find the initial and final temperatures:
T_1 = \frac{P_1 V_1}{nR}
T_2 = \frac{P_2 V_2}{nR} = \frac{\frac{P_1}{4} \cdot 2V_1}{nR} = \frac{P_1 V_1}{2nR} = \frac{T_1}{2}
This indicates that the temperature decreases:
ΔT = T_2 - T_1 = \frac{T_1}{2} - T_1 = -\frac{T_1}{2}
Heat Change
For an ideal gas, the heat change can be calculated using:
Q = nC_vΔT
For a monoatomic gas, the molar heat capacity at constant volume \( C_v = \frac{3R}{2} \). Thus:
Q = n \cdot \frac{3R}{2} \cdot \left(-\frac{T_1}{2}\right) = -\frac{3nRT_1}{4}
This indicates a heat loss.
Case 2: Expansion with \( PV^{4/3} = \text{constant} \)
In this case, we have:
P_1 V_1^{4/3} = P_2 V_2^{4/3}
Again, substituting \( V_2 = 2V_1 \):
P_1 V_1^{4/3} = P_2 (2V_1)^{4/3}
This simplifies to:
P_1 V_1^{4/3} = P_2 \cdot 2^{4/3} V_1^{4/3}
From this, we can derive:
P_2 = \frac{P_1}{2^{4/3}}
Temperature Change
Using the ideal gas law again:
T_2 = \frac{P_2 V_2}{nR} = \frac{\frac{P_1}{2^{4/3}} \cdot 2V_1}{nR} = \frac{P_1 V_1}{2^{1/3} nR} = 2^{2/3} T_1
Thus, the temperature increases:
ΔT = T_2 - T_1 = 2^{2/3} T_1 - T_1 = (2^{2/3} - 1)T_1
Heat Change
Calculating the heat change for this case:
Q = nC_vΔT = n \cdot \frac{3R}{2} \cdot (2^{2/3} - 1)T_1
This indicates a heat gain since \( 2^{2/3} - 1 \) is positive.
Summary of Results
In summary:
- For \( PV^2 = \text{constant} \): Temperature decreases and heat is lost.
- For \( PV^{4/3} = \text{constant} \): Temperature increases and heat is gained.
This analysis illustrates how different constraints on the gas's expansion can lead to significantly different thermal behaviors. Understanding these principles is crucial for applications in thermodynamics and engineering.
