Short Notes: Thermodynamics
First Law of Thermodynamics
The First Law of Thermodynamics is a statement of conservation of energy for thermodynamic systems. It relates the heat added to a system, the change in its internal energy and the work done by (or on) the system. The law applies to macroscopic systems and is valid for reversible and irreversible processes alike.
The law, in differential form for a closed system, may be written as:
\( \Delta Q = \Delta U + \Delta W \)
| Concept | Formula / Notes |
|---|---|
| First Law | \( \Delta Q = \Delta U + \Delta W \) |
| Work done by a gas (constant pressure) | \( W = P\,\Delta V \) |
| Change of internal energy (ideal gas) | \( \Delta U = n C_{V} \Delta T \) |
Sign convention commonly used here is that heat added to the system is positive and work done by the system is positive. For processes where work is done on the system the work term becomes negative.
Thermodynamic Processes
A thermodynamic process describes how a system changes state. The following processes are commonly encountered for ideal gases. Each is defined by a constraint that remains constant during the process.
| Process | Characteristics / Relations |
|---|---|
| Isothermal | \( \Delta T = 0; \;\Delta U = 0; \;Q = W; \;PV = \text{constant} \) |
| Adiabatic | \( Q = 0; \;\Delta U = -W; \;PV^{\gamma} = \text{constant}; \;TV^{\gamma-1} = \text{constant} \) |
| Isobaric | \( P = \text{constant}; \;W = P\Delta V; \;Q = n C_{P} \Delta T \) |
| Isochoric | \( V = \text{constant}; \;W = 0; \;Q = \Delta U = n C_{V} \Delta T \) |
Physical notes and examples:
- Isothermal process: Temperature remains constant. Heat exchanged equals work done. Example: slow compression/expansion of a gas while the cylinder is kept in contact with a heat reservoir.
- Adiabatic process: No heat exchange with surroundings (perfect insulation). The temperature of the gas changes due to work done. Example: rapid compression in a well-insulated piston.
- Isobaric process: Pressure remains constant. Heating at constant pressure causes volume to change. Example: heating a gas in a cylinder with a freely moving piston supporting a constant external pressure.
- Isochoric process: Volume fixed, so no boundary work. All heat goes into changing internal energy. Example: heating gas in a rigid container.
Specific Heat Relations
Specific heats at constant pressure and volume relate heat supplied to the temperature rise for a given amount of substance. For an ideal gas they are linked by Mayer's relation and the heat-capacity ratio.
| Relation | Formula / Values |
|---|---|
| Mayer's relation | \( C_{P} - C_{V} = R \) |
| Heat capacity ratio (gamma) | \( \gamma = \dfrac{C_{P}}{C_{V}} \) |
| Monoatomic ideal gas | \( C_{V} = \tfrac{3}{2}R; \; C_{P} = \tfrac{5}{2}R; \; \gamma = \tfrac{5}{3} \) |
| Diatomic ideal gas (at ordinary temperatures) | \( C_{V} = \tfrac{5}{2}R; \; C_{P} = \tfrac{7}{2}R; \; \gamma = \tfrac{7}{5} \) |
| Polyatomic (approx.) | \( C_{V} \approx 3R; \; C_{P} \approx 4R; \; \gamma \approx \tfrac{4}{3} \) |
These values follow from the equipartition theorem where each translational or rotational degree of freedom contributes \(\tfrac{1}{2}R\) per mole to the molar heat capacity at constant volume.
Work Done in Different Processes
Work done by a gas in a quasi-static process is the area under the pressure-volume curve. Common formulae for ideal-gas processes:
| Process | Work Formula |
|---|---|
| Isothermal (ideal gas) | \( W = nRT \ln\!\dfrac{V_{2}}{V_{1}} = nRT \ln\!\dfrac{P_{1}}{P_{2}} \) |
| Adiabatic | \( W = \dfrac{P_{1}V_{1} - P_{2}V_{2}}{\gamma - 1} = n C_{V} (T_{1} - T_{2}) \) |
| Isobaric | \( W = P (V_{2} - V_{1}) = n R \Delta T \) |
| Isochoric | \( W = 0 \) |
Brief derivations and remarks:
- For an isothermal process of an ideal gas, pressure varies as \(P = \dfrac{nRT}{V}\). Work is the integral \( \int_{V_{1}}^{V_{2}} P\,dV \), which gives the logarithmic expression above.
- For an adiabatic quasi-static process, use \(PV^{\gamma}=\) constant and the first law with \(Q=0\) to obtain the stated expressions. The sign of \(W\) depends on whether the gas expands or is compressed.
- For an isobaric process, pressure is constant and the boundary work equals \(P\Delta V\).
- For an isochoric process, volume is fixed so boundary work is zero and heat changes internal energy only.
Heat Engines and Refrigerators
A heat engine converts heat absorbed from a high-temperature reservoir into work while discarding some heat to a low-temperature reservoir. A refrigerator or heat pump uses work to transfer heat from a colder body to a hotter body.
| Concept | Formula |
|---|---|
| Efficiency of an engine | \( \eta = \dfrac{W}{Q_{1}} = \dfrac{Q_{1} - Q_{2}}{Q_{1}} = 1 - \dfrac{Q_{2}}{Q_{1}} \) |
| Carnot efficiency (reversible engine) | \( \eta_{\text{Carnot}} = 1 - \dfrac{T_{2}}{T_{1}} = \dfrac{T_{1} - T_{2}}{T_{1}} \) |
| Coefficient of performance (refrigerator) | \( \beta = \dfrac{Q_{2}}{W} = \dfrac{Q_{2}}{Q_{1}-Q_{2}} = \dfrac{T_{2}}{T_{1}-T_{2}} \) |
Remarks:
- Efficiency measures the fraction of heat input converted to useful work. No real engine can exceed the Carnot efficiency between the same two temperatures.
- Carnot engine is an ideal reversible engine operating between two reservoirs at temperatures \(T_{1}\) (hot) and \(T_{2}\) (cold). Its efficiency depends only on these temperatures and not on the working substance.
- Coefficient of performance for refrigerators compares heat removed from the cold space with the work input. A larger COP means a more effective refrigerator for given temperatures.
Second Law of Thermodynamics
The Second Law of Thermodynamics introduces the concept of irreversibility and sets limits on the direction of natural processes. It is expressed in several equivalent ways.
- Kelvin-Planck statement: It is impossible to construct an engine operating in a cycle whose sole effect is to convert all heat absorbed from a single heat reservoir into work. In other words, no heat engine can have 100% efficiency when operating between a single reservoir and producing net work.
- Clausius statement: Heat cannot spontaneously flow from a colder body to a hotter body without the input of external work.
- Carnot theorem: No engine operating between two heat reservoirs can be more efficient than a reversible (Carnot) engine operating between the same reservoirs.
- Reversibility: All reversible engines operating between the same two temperatures have the same efficiency. Irreversible processes have strictly lower efficiencies.
Entropy provides a quantitative measure of irreversibility. For a reversible change in which an amount of heat \(dQ_{\text{rev}}\) is absorbed at temperature \(T\), the change in entropy is:
\( dS = \dfrac{dQ_{\text{rev}}}{T} \)
For an isolated system, the total entropy cannot decrease; for irreversible processes entropy increases. This underlies the directionality of natural processes and explains why some cyclic processes cannot be run in reverse without external work.
Useful Keywords and Formula Summary
- First Law: \( \Delta Q = \Delta U + \Delta W \)
- Work (constant P): \( W = P \Delta V \)
- Internal energy (ideal gas): \( \Delta U = n C_{V} \Delta T \)
- Mayer's relation: \( C_{P} - C_{V} = R \)
- Heat capacity ratio: \( \gamma = \dfrac{C_{P}}{C_{V}} \)
- Isothermal work: \( W = nRT \ln\dfrac{V_{2}}{V_{1}} \)
- Adiabatic relation: \( PV^{\gamma} = \text{constant} \)
- Efficiency: \( \eta = 1 - \dfrac{Q_{2}}{Q_{1}} = 1 - \dfrac{T_{2}}{T_{1}} \) for Carnot
- Entropy: \( dS = \dfrac{dQ_{\text{rev}}}{T} \)
FAQs on Short Notes: Thermodynamics
| 1. What is thermodynamics? |  |
| 2. What are the laws of thermodynamics? | |
| 3. What is the difference between an open, closed, and isolated system in thermodynamics? | |
| 4. What is the significance of the concept of entropy in thermodynamics? | |
| 5. How is the concept of heat capacity important in thermodynamics? | |
