An isothermal process is a thermodynamic process that occurs at a constant temperature. In such a process, the internal energy of an ideal gas remains constant because temperature is directly related to internal energy. The heat added to the system is equal to the work done by the system.
For an ideal gas undergoing an isothermal process, the relationship can be described by the equation:
\[ PV = nRT \]
Where:
- \(P\) = Pressure of the gas
- \(V\) = Volume of the gas
- \(n\) = Number of moles of the gas
- \(R\) = Universal gas constant
- \(T\) = Absolute temperature (in Kelvin)
In an isothermal expansion or compression, the product \(PV\) remains constant because \(T\) is constant. Thus, if the volume increases, the pressure decreases, and vice versa, while maintaining the equality \(P_1V_1 = P_2V_2\).
The work done \(W\) during an isothermal process can be calculated using:
\[ W = nRT \ln\left(\frac{V_f}{V_i}\right) \]
Where:
- \(W\) = Work done by the gas
- \(V_f\) = Final volume
- \(V_i\) = Initial volume
Here, the logarithmic term accounts for the changing volume during the process, and \(W\) is positive if work is done by the gas when it expands.
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