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 consider the two different steps: heating at constant volume and then heating at constant pressure.
Step 1: Heating at constant volume
In this step, the volume remains constant while the gas is heated until its pressure becomes 3 times the initial pressure. Since the volume is constant, the work done by the gas is zero. Therefore, the heat added to the system equals the change in internal energy.
We can calculate the change in internal energy (ΔU) using the First Law of Thermodynamics equation:
ΔU = Q - W
Since there is no work done (W = 0), ΔU = Q.
The change in internal energy (ΔU) can be expressed as the product of the molar heat capacity at constant volume (Cv) and the change in temperature (ΔT):
ΔU = Cv * ΔT
Therefore, in this step, the heat added to the system (Q1) is given by:
Q1 = Cv * ΔT1,
where ΔT1 is the change in temperature in step 1.
Step 2: Heating at constant pressure
In this step, the volume is doubled while the gas is heated at constant pressure. Here, both heat (Q2) and work (W) are involved. The change in internal energy (ΔU) can be given as:
ΔU = Q2 - W
Since the volume is doubled at constant pressure, the work done by the gas is given by W = PΔV. Here, P is the constant pressure and ΔV is the change in volume.
The heat added to the system (Q2) is also given by:
Q2 = Cp * ΔT2,
where Cp is the molar heat capacity at constant pressure, and ΔT2 is the change in temperature in step 2.
Since the volume is doubled, ΔV = V, where V is the initial volume of the gas.
Now, to find the molar heat capacity for the whole process, we need to add up the heat added in both steps:
Q_total = Q1 + Q2
To find the molar heat capacity (Cp_total) for the whole process, we divide the total heat added by the change in temperature (ΔT_total):
Cp_total = Q_total / ΔT_total
Finally, we can express the molar heat capacity for the whole process as a sum of two terms:
Cp_total = (Cv * ΔT1 + Cp * ΔT2) / ΔT_total
Note: The values for ΔT1, ΔT2, Cv, and Cp need to be provided in order to calculate the molar heat capacity for the whole process.