The relationship between volume and internal energy in reversible adiabatic expansion
- Reversible adiabatic expansion:
In reversible adiabatic expansion, the process occurs without the transfer of heat to or from the surroundings, and the system is thermally isolated. This means that the change in internal energy of the gas is solely determined by the work done on or by the gas.
- First law of thermodynamics:
The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In the case of adiabatic expansion, where Q = 0, the change in internal energy is solely due to work done.
- Relationship between volume and internal energy:
For a perfect mono-atomic gas undergoing reversible adiabatic expansion, the relationship between volume (V) and internal energy (U) can be expressed as:
\[ U = C_v \times T = \frac{3}{2}nRT \]
where:
- U is the internal energy
- Cv is the molar specific heat at constant volume
- T is the temperature
- n is the number of moles of gas
- R is the ideal gas constant
- Explanation:
During adiabatic expansion, the internal energy of the gas changes due to the work done on or by the gas. Since the process is reversible, the gas remains in thermal equilibrium throughout the expansion. The internal energy of the gas is related to its temperature, which in turn is related to the volume of the gas through the ideal gas law.
- Conclusion:
In summary, the relationship between volume and internal energy for a perfect mono-atomic gas undergoing reversible adiabatic expansion is determined by the first law of thermodynamics and the ideal gas law. The change in internal energy is solely due to the work done during the expansion, and this change is reflected in the temperature and volume of the gas.