# Applications of the First Law of Thermodynamics The **First Law of Thermodynamics** ($Q = \Delta U + W$) is a universal principle of energy conservation. Its application becomes clearer when we examine specific thermodynamic processes where one variable such as volume, pressure, temperature, or heat is held constant. ## 1. Isochoric Process (Constant Volume) An isochoric process is one where the volume of the system does not change. **Condition:** $\Delta V = 0$ **Work Done:** Since work is defined as $W = P\Delta V$, no work is done by or on the system. $$W = 0$$ **First Law Application:** With $W = 0$, the First Law simplifies to: $$Q = \Delta U$$ This means that **all heat added to the system goes directly into increasing its internal energy**, which typically results in an increase in its temperature and pressure. **PV Graph:** An isochoric process is represented by a **vertical line** on a pressure-volume (PV) diagram. ## 2. Isobaric Process (Constant Pressure) An isobaric process is one where the pressure of the system remains constant. **Condition:** Pressure $P$ is constant. **Work Done:** As the volume changes from $V_1$ to $V_2$, the work done is: $$W = P \Delta V = P(V_2 - V_1)$$ **First Law Application:** The First Law remains in its full form: $$Q = \Delta U + P\Delta V$$ This means the heat added to the system is used for both **increasing the internal energy** and **doing work** on the surroundings (if it expands). **PV Graph:** An isobaric process is represented by a **horizontal line** on a PV diagram. ## 3. Isothermal Process (Constant Temperature) An isothermal process is one where the temperature of the system remains constant. For an ideal gas, the internal energy depends only on temperature. **Condition:** $\Delta T = 0$ **Internal Energy:** For an ideal gas, the internal energy is a function of temperature only. Therefore, the change in internal energy is zero. $$\Delta U = 0$$ **First Law Application:** With $\Delta U = 0$, the First Law simplifies to: $$Q = W$$ This means **all heat added to the system is converted into work done by the system**. To keep the temperature constant during an expansion, heat must be supplied to the system. **PV Graph:** An isothermal process is represented by a **hyperbolic curve** on a PV diagram, following Boyle's Law ($PV = \text{constant}$). ## 4. Adiabatic Process (No Heat Exchange) An adiabatic process is one where no heat enters or leaves the system. **Condition:** $Q = 0$ **First Law Application:** With $Q = 0$, the First Law becomes: $$0 = \Delta U + W \quad \implies \quad \Delta U = -W$$ This means **any work done by the system comes at the expense of its internal energy**. During an adiabatic expansion, the system does work, its internal energy decreases, and its temperature drops. **PV Graph:** An adiabatic process is also a curve, but it is **steeper than an isothermal curve** because the temperature drops during expansion, causing the pressure to fall more rapidly. ### Comparison: Adiabatic vs. Isothermal Expansion Starting from the same point, an adiabatic expansion results in a lower final pressure and temperature compared to an isothermal expansion to the same final volume. This is because, in the adiabatic case, the internal energy drops, while in the isothermal case, heat is added to keep the temperature (and thus internal energy) constant. ## Summary Table | Process | Condition | First Law Equation | Key Characteristic | | :--- | :--- | :--- | :--- | | **Isochoric** | Constant Volume ($\Delta V = 0$) | $Q = \Delta U$ | No work is done ($W = 0$). | | **Isobaric** | Constant Pressure | $Q = \Delta U + P\Delta V$ | Heat is used to change internal energy and do work. | | **Isothermal** | Constant Temperature ($\Delta T = 0$) | $Q = W$ | No change in internal energy ($\Delta U = 0$). | | **Adiabatic** | No Heat Exchange ($Q = 0$) | $\Delta U = -W$ | Work is done at the expense of internal energy. | ## Reference