An isothermal process is a thermodynamic process that occurs at a constant temperature. In this process, the internal energy of an ideal gas remains constant, which means that any heat added to the system is used to do work by the system, and vice versa.
The key equation describing an isothermal process for an ideal gas is derived from the ideal gas law and the first law of thermodynamics. For an ideal gas, the relationship between pressure (P), volume (V), and temperature (T) is given by:
\[ PV = nRT \]
where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles of gas,
- \( R \) is the ideal gas constant,
- \( T \) is the absolute temperature.
During an isothermal expansion or compression, the product \( PV \) remains constant (since \( T \) is constant). Thus, we have:
\[ P_1 V_1 = P_2 V_2 \]
The work done (W) by or on the gas during an isothermal process can be calculated using:
\[ W = nRT \ln \left( \frac{V_f}{V_i} \right) \]
where:
- \( V_f \) is the final volume,
- \( V_i \) is the initial volume.
In summary, an isothermal process is characterized by constant temperature, and the work done during this process can be calculated using the above equations.