Ratio of Cp and Cv:
The ratio of the molar specific heat of a gas at constant pressure (Cp) to the molar specific heat at constant volume (Cv) is known as the heat capacity ratio or the adiabatic index of a gas and is denoted by the symbol "γ". The value of γ is greater than one, and it depends on the temperature and pressure of the gas.
Derivation of Relation: Cp - Cv = R:
The relation between the molar specific heat at constant pressure (Cp) and the molar specific heat at constant volume (Cv) can be derived using the first law of thermodynamics. According to this law, the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.
- For a gas, the internal energy of the system is given by the equation:
- U = Cv * T
- Where U is the internal energy, Cv is the molar specific heat at constant volume, and T is the temperature.
- The heat added to the system is given by the equation:
- Q = Cp * T
- Where Q is the heat added, Cp is the molar specific heat at constant pressure, and T is the temperature.
- The work done by the system is given by the equation:
- W = P * ΔV
- Where W is the work done, P is the pressure, and ΔV is the change in volume.
- Using the first law of thermodynamics, the change in internal energy is given by the equation:
- ΔU = Q - W
- Substituting the values of Q and W from the above equations, we get:
- ΔU = Cp * T - P * ΔV
- Now, substituting the value of U from the first equation, we get:
- Cv * ΔT = Cp * ΔT - P * ΔV
- Dividing both sides of the equation by ΔT, we get:
- Cv = Cp - P * ΔV / ΔT
- Finally, dividing both sides of the equation by ΔV, we get:
- Cv = Cp - R
Where R is the gas constant.
In conclusion, the relation between the molar specific heat at constant pressure (Cp) and the molar specific heat at constant volume (Cv) is given by the equation:
Cp - Cv = R.