A diatomic ideal gas is heated at constant volume until its pressure becomes 3 times.it is again heated at constant pressure until its volume is doubled.find the molar heat capacity for the whole process?

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To find the molar heat capacity for the whole process, we need to consider both the constant volume and constant pressure stages separately. The molar heat capacity at constant volume (Cv) and constant pressure (Cp) are significant to calculate the overall molar heat capacity (C).

1. Constant Volume Stage:
In this stage, the gas is heated at constant volume until its pressure becomes 3 times. Let's assume the initial pressure and volume are P1 and V1, respectively. After heating, the final pressure becomes 3P1.

We can use the relationship between pressure and temperature for an ideal gas:
P1V1 / T1 = (3P1)(V1) / T2

Notice that the volume cancels out, leaving us with:
T1 / T2 = 1/3

Since the process is taking place at constant volume, we can use the molar heat capacity at constant volume (Cv) to find the heat (Q1) absorbed:
Q1 = nCvΔTv

Here, ΔTv represents the change in temperature at constant volume, given by T2 - T1.

2. Constant Pressure Stage:
In this stage, the gas is heated at constant pressure until its volume is doubled. Let's assume the initial volume and pressure are V1 and P1, respectively. After heating, the final volume becomes 2V1.

Again, using the relationship between pressure and temperature for an ideal gas:
P1V1 / T2 = P1(2V1) / T3

Simplifying this equation, we find:
T2 / T3 = 1/2

Since the process is taking place at constant pressure, we can use the molar heat capacity at constant pressure (Cp) to find the heat (Q2) absorbed:
Q2 = nCpΔTp

Here, ΔTp represents the change in temperature at constant pressure, given by T3 - T2.

3. Overall Molar Heat Capacity:
The overall molar heat capacity (C) is calculated as the sum of the molar heat capacities at constant volume (Cv) and constant pressure (Cp):
C = Cv + Cp

Now that we have obtained Q1, Q2, and ΔT, we can calculate the molar heat capacities at constant volume (Cv) and constant pressure (Cp) using the formulas mentioned above. Finally, add Cv and Cp to find the overall molar heat capacity (C).