- Thermodynamics explores how energy is transferred and transformed, particularly through heat and work.
- These transfers are governed by the first law of thermodynamics, which is a specific application of the law of conservation of energy.
Energy can be transferred or transformed, but it cannot be created or destroyed.
The first law applies the conservation of energy to thermodynamic systems.
Energy conservation in a thermodynamic system involves three key quantities:
- Internal energy ($U$): the total energy of the particles in a system, made up of their random kinetic and potential energy.
- Heat ($Q$): energy transferred between a system and its surroundings because of a temperature difference.
- Work ($W$): energy transferred when a force acts over a distance, such as when a gas expands or is compressed.
Using the Clausius sign convention, the first law is written as:
$$Q = \Delta U + W$$
where:
- $Q$ is the heat supplied to the system.
- $\Delta U$ is the change in internal energy of the system.
- $W$ is the work done by the system.
The sign convention is important:
- $Q > 0$: heat is added to the system.
- $Q < 0$: heat is removed from the system.
- $W > 0$: work is done by the system (e.g., expansion).
- $W < 0$: work is done on the system (e.g., compression).
- For a monatomic ideal gas, the internal energy is entirely the random translational kinetic energy of its particles.
- Its change is: $$\Delta U = \tfrac{3}{2}Nk_B\Delta T = \tfrac{3}{2}nR\Delta T$$ where:
- $N$ is the number of particles and $k_B$ is the Boltzmann constant.
- $n$ is the number of moles and $R$ is the gas constant.
- The internal energy of an ideal gas depends only on its temperature.
- When the temperature does not change, $\Delta U = 0$.
The equation $Q = \Delta U + W$ shows that heat supplied to a system is used to either:
- increase the system's internal energy ($\Delta U$), or
- do work ($W$) on the surroundings.
If no heat is added or removed ($Q = 0$), any work done by the system must come from its internal energy, so $U$ decreases.
- Consider a gas in a cylinder fitted with a piston.
- Heating the gas makes it expand and push the piston outward.
- The heat supplied both increases the gas's internal energy and does work on the piston.
In thermodynamics, work is usually associated with a change in the volume of a gas.
- When a gas expands or is compressed, work is done.
- At constant external pressure, the work done by the gas is $$W = P_{\text{ext}} \Delta V$$ where:
- $W$ is the work done by the gas.
- $P_{\text{ext}}$ is the external pressure (assumed constant during the process).
- $\Delta V$ is the change in volume of the gas.
- If the gas expands, $\Delta V > 0$ and $W > 0$ (work is done by the gas).
- If the gas is compressed, $\Delta V < 0$ and $W < 0$ (work is done on the gas).
If the pressure is not constant, the work done is the area under the curve on a pressure-volume ($P$-$V$) diagram.
- A gas expands at a constant pressure of $2.0 \times 10^5 \, \text{Pa}$.
- If the volume increases from $1.0 \, \text{m}^3$ to $1.5 \, \text{m}^3$, the work done by the gas is:
$$W = P_{\text{ext}} \Delta V = 2.0 \times 10^5 \, \text{Pa} \times (1.5 - 1.0) \, \text{m}^3 = 1.0 \times 10^5 \, \text{J}$$
- When the pressure is not constant, divide the $P$-$V$ curve into small segments where the pressure is approximately constant.
- Calculate the work for each segment and sum the results.
- The first law takes a simpler form in each of the standard processes, depending on which quantity is held constant.
- The isovolumetric, isobaric, isothermal, and adiabatic processes are analysed in detail in B.4.3 Thermodynamic processes and heat engines.
- Students often confuse the signs of $Q$ and $W$.
- Positive $Q$ means heat is added to the system.
- Positive $W$ means work is done by the system.
- State the first law of thermodynamics and define each term in $Q = \Delta U + W$.
- In the Clausius convention, what do positive $Q$ and positive $W$ each mean?
- Why does the internal energy of an ideal gas depend only on temperature?
- How do you find the work done by a gas from a $P$-$V$ diagram when pressure is not constant?