To determine the relationship between the pressures and temperatures of the gas in states A and B, we need to analyze the given information about the specific heat capacities at constant pressure (Cp) and constant volume (Cv). The difference between these two values, known as the gas constant (R), plays a crucial role in understanding the behavior of the gas under different conditions.
Understanding Cp and Cv
The specific heat capacities are defined as follows:
- Cp: The amount of heat required to raise the temperature of a unit mass of gas at constant pressure.
- Cv: The amount of heat required to raise the temperature of a unit mass of gas at constant volume.
The relationship between Cp and Cv for an ideal gas is given by the equation:
Cp - Cv = R
Here, R is the universal gas constant. In your case, the values of Cp - Cv are provided for two states:
- For state A: Cp - Cv = 1.0 R
- For state B: Cp - Cv = 1.08 R
Analyzing the States
From the above, we can conclude that:
- In state A, the gas behaves with a specific heat difference of 1.0 R.
- In state B, the gas has a slightly higher specific heat difference of 1.08 R.
This indicates that the gas in state B has a higher capacity to store heat at constant pressure compared to state A. This could imply that the gas in state B is at a higher temperature or lower pressure than in state A, as higher temperatures generally lead to higher Cp values.
Applying the Ideal Gas Law
The ideal gas law states:
P * V = n * R * T
Where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. If we assume the volume and the number of moles remain constant between the two states, we can analyze the relationship between pressure and temperature:
- If the temperature increases (Ta > Tb), then according to the ideal gas law, the pressure must decrease (Pa < Pb) to maintain the equality.
- If the temperature decreases (Ta < Tb), then the pressure must increase (Pa > Pb).
Conclusion from Analysis
Given that in state A, the specific heat difference is less than in state B, it suggests that state B is at a higher temperature. Therefore, we can conclude:
- Since Ta > Tb, it follows that Pa < Pb.
Thus, the correct answer to your question is (a) Pa < Pb and Ta > Tb.
