Problem Statement
Show that the heat absorbed in a reversible isobaric process equals the enthalpy change: $Q_p = \Delta H$.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: Enthalpy is defined as $H = U + pV$. At constant pressure:
Step 2 — Apply the relevant physical law or equation: $$dH = dU + p\,dV + V\,dp = dU + p\,dV\quad (dp=0)$$
Step 3 — Solve algebraically for the unknown: By the first law: $dQ = dU + p\,dV$. Therefore:
Step 4 — Substitute numerical values with units: $$dQ_p = dH \implies Q_p = \Delta H$$
Step 5 — Compute and check the result: This is why $C_p = (\partial H/\partial T)_p$ for an ideal gas. Enthalpy is the natural thermodynamic potential for constant-pressure processes (such as most chemical reactions).
Worked Calculation
$$dH = dU + p\,dV + V\,dp = dU + p\,dV\quad (dp=0)$$
$$dQ_p = dH \implies Q_p = \Delta H$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
Enthalpy is defined as $H = U + pV$. At constant pressure:
$$dH = dU + p\,dV + V\,dp = dU + p\,dV\quad (dp=0)$$
By the first law: $dQ = dU + p\,dV$. Therefore:
$$dQ_p = dH \implies Q_p = \Delta H$$
This is why $C_p = (\partial H/\partial T)_p$ for an ideal gas. Enthalpy is the natural thermodynamic potential for constant-pressure processes (such as most chemical reactions).
Answer
$$\boxed{dQ_p = dH \implies Q_p = \Delta H}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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