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?
To find the molar heat capacity for the whole process, we need to calculate the heat absorbed by the gas during each step of the process.
Step 1: Heating at constant volume
First, let's consider the initial and final states of the gas in this step. Let the initial temperature, pressure, and volume be T1, P1, and V1, respectively. The final pressure is given as 3P1.
Since the process is taking place at constant volume, we can use the equation:
Q1 = n * Cv * ΔT
where Q1 is the heat absorbed, n is the number of moles of gas, Cv is the molar heat capacity at constant volume, and ΔT is the change in temperature. Here, since the volume is constant, ΔT = T2 - T1, where T2 is the final temperature.
Step 2: Heating at constant pressure
In this step, the volume of the gas is doubled. Let's consider the final volume after doubling as 2V1.
Since the process is taking place at constant pressure, we can use the equation:
Q2 = n * Cp * ΔT
where Q2 is the heat absorbed, n is the number of moles of gas, Cp is the molar heat capacity at constant pressure, and ΔT is the change in temperature. Here, since the pressure is constant, ΔT = T3 - T2, where T3 is the final temperature.
Now, to find the molar heat capacity for the whole process, we need to sum the heat absorbed in both steps:
Q_total = Q1 + Q2
Since Cv and Cp are defined as the amount of heat required to raise the temperature of one mole of gas by 1 degree Celsius at constant volume and pressure, respectively, we can write:
C_total = (Q1 + Q2) / (n * ΔT)
Substituting the expressions for Q1 and Q2 from the above equations, we get:
C_total = (n * Cv * ΔT + n * Cp * ΔT) / (n * ΔT)
Simplifying the equation, we find:
C_total = Cv + Cp
Therefore, the molar heat capacity for the whole process is equal to the sum of the molar heat capacities at constant volume and constant pressure.
In summary, to find the molar heat capacity for the whole process of heating a diatomic ideal gas at constant volume and then at constant pressure, you simply add the molar heat capacities at constant volume and constant pressure.