Isothermal process
Isothermal Process
An isothermal process is a thermodynamic process in which the temperature of the system remains constant throughout. The term "isothermal" comes from the Greek words "iso" (equal) and "therme" (heat), indicating that the temperature does not change during the process. This is a key concept in the study of thermodynamics, particularly when dealing with ideal gases.
Understanding Isothermal Processes
In an isothermal process, the system is in thermal equilibrium with its surroundings or a heat reservoir. Heat transfer occurs between the system and the surroundings in such a way that the temperature of the system remains constant. For an ideal gas, this implies that the product of pressure (P) and volume (V) remains constant, as described by Boyle's Law.
Thermodynamic Description
The first law of thermodynamics states that the change in internal energy (∆U) of a system is equal to the heat added to the system (Q) minus the work done by the system (W):
[ \Delta U = Q - W ]
For an isothermal process involving an ideal gas, the internal energy depends only on temperature. Since the temperature is constant, the change in internal energy is zero (∆U = 0). Therefore, the heat added to the system is equal to the work done by the system:
[ Q = W ]
Mathematical Representation
For an ideal gas undergoing an isothermal process, Boyle's Law can be applied:
[ PV = \text{constant} ]
If the initial state of the gas is given by ( P_1, V_1 ) and the final state by ( P_2, V_2 ), then:
[ P_1V_1 = P_2V_2 ]
The work done by the gas during an isothermal expansion or compression can be calculated by integrating the pressure with respect to volume:
[ W = \int_{V_1}^{V_2} P \, dV ]
Since ( P = \frac{nRT}{V} ) for an ideal gas (where n is the number of moles, R is the ideal gas constant, and T is the temperature), the work done in an isothermal process is:
[ W = nRT \ln\left(\frac{V_2}{V_1}\right) ]
Differences and Important Points
Here is a table summarizing the key differences and important points of an isothermal process compared to other thermodynamic processes:
| Feature | Isothermal Process | Other Processes |
|---|---|---|
| Temperature | Constant | May change |
| Internal Energy Change (∆U) | Zero | Depends on temperature change |
| Heat Transfer (Q) | Equal to work done (W) | May not be equal to W |
| Work Done (W) | Depends on volume change | Depends on specific process |
| Ideal Gas Law | ( PV = \text{constant} ) | May not hold |
| Path on P-V Diagram | Hyperbolic curve | Varies (e.g., straight line for isobaric) |
Examples
Example 1: Isothermal Expansion of an Ideal Gas
Consider 1 mole of an ideal gas that expands isothermally from a volume of 1 L to 2 L at a constant temperature of 300 K. The work done by the gas can be calculated as follows:
Given:
- ( n = 1 ) mole
- ( R = 8.314 \, \text{J/(mol·K)} )
- ( T = 300 \, \text{K} )
- ( V_1 = 1 \, \text{L} )
- ( V_2 = 2 \, \text{L} )
[ W = nRT \ln\left(\frac{V_2}{V_1}\right) ] [ W = (1 \, \text{mol})(8.314 \, \text{J/(mol·K)})(300 \, \text{K}) \ln\left(\frac{2 \, \text{L}}{1 \, \text{L}}\right) ] [ W \approx 1727 \, \text{J} ]
Example 2: Isothermal Compression of an Ideal Gas
If the same ideal gas is isothermally compressed back to its original volume (from 2 L to 1 L), the work done on the gas is:
[ W = nRT \ln\left(\frac{V_1}{V_2}\right) ] [ W = (1 \, \text{mol})(8.314 \, \text{J/(mol·K)})(300 \, \text{K}) \ln\left(\frac{1 \, \text{L}}{2 \, \text{L}}\right) ] [ W \approx -1727 \, \text{J} ]
The negative sign indicates that work is done on the gas, as opposed to by the gas.
In conclusion, an isothermal process is characterized by constant temperature and a direct relationship between heat transfer and work done. It is an important concept in thermodynamics, particularly in the study of ideal gases and heat engines.