Q. \(\Delta E = q + w\).

Step 1. State the law and correct the notation. The first law of thermodynamics relates the change in internal energy to heat and work. Written with standard LaTeX notation, the law is

\[ \Delta E = q + w. \]

Here \( \Delta E \) is the change in the internal energy of the system, \( q \) is the heat added to the system, and \( w \) is the work done on the system. This is an energy balance: energy change equals energy in by heat plus energy in by work.

Step 2. Explain sign conventions. According to the sign convention used above, positive \( q \) means heat flows into the system. Positive \( w \) means work is done on the system. With this convention, if the system does work on the surroundings, \( w \) is negative. If heat is lost by the system, \( q \) is negative.

Step 3. Differential / path notation and exactness. Heat and work are path functions, not state functions. To emphasize this, their differentials are commonly written with a partial (inexact) symbol. In differential form the first law is

\[ dE = \delta q + \delta w. \]

Here \( dE \) is an exact differential because internal energy is a state function, while \( \delta q \) and \( \delta w \) are inexact differentials because heat and work depend on the process path.

Step 4. Typical form of mechanical (pressure–volume) work. For a closed system where the only work mode is pressure–volume work against an external pressure \( P_{\text{ext}} \), the work done on the system during a volume change from \( V_1 \) to \( V_2 \) is

\[ w = -\int_{V_1}^{V_2} P_{\text{ext}}\, dV. \]

With the sign convention above, if the system expands (so \( V_2 > V_1 \) and \( dV > 0 \)), then the integral is positive and \( w \) is negative, indicating net work done by the system on the surroundings.

Step 5. How to use the equation in practice. Procedure:

1. Identify the system and determine which energies cross the system boundary as heat and which as work. 2. Use the appropriate expression for work (for example, \( w = -\int P_{\text{ext}}\, dV \) for PV work). 3. Evaluate or measure \( q \) and \( w \) for the process. 4. Compute the change in internal energy from \( \Delta E = q + w \).

Step 6. Short numerical example. Suppose a system absorbs \( 200\ \mathrm{J} \) of heat and does \( 50\ \mathrm{J} \) of work on the surroundings. With our sign convention, \( q = +200\ \mathrm{J} \) and \( w = -50\ \mathrm{J} \). Then

\[ \Delta E = q + w = 200\ \mathrm{J} + (-50\ \mathrm{J}) = 150\ \mathrm{J}. \]

Thus the internal energy of the system increases by \( 150\ \mathrm{J} \).

Step 7. Key conceptual points. (i) \( \Delta E \) depends only on initial and final states. (ii) \( q \) and \( w \) are path dependent. (iii) Energy is conserved: any change in internal energy must be accounted for by heat and/or work exchanged.