Derive a relationship between ΔH and ΔE for a process in which the temperature of a fixed amount of an ideal gas changes.

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1. Start with the first law of thermodynamics: \( \Delta E = q + w \), where \( \Delta E \) is the change in internal energy, \( q \) is the heat exchanged, and \( w \) is the work done.
2. For an ideal gas, the work done by the system during expansion or compression at constant pressure is given by \( w = -P_{\text{ext}} \Delta V \), where \( P_{\text{ext}} \) is the external pressure and \( \Delta V \) is the change in volume.
3. The enthalpy change \( \Delta H \) is defined as \( \Delta H = \Delta E + P \Delta V \).
4. Substitute the expression for work into the first law equation: \( \Delta E = q - P_{\text{ext}} \Delta V \).
5. Combine the expressions for \( \Delta E \) and \( \Delta H \) to derive the relationship: \( \Delta H = q - P_{\text{ext}} \Delta V + P \Delta V = q + (P - P_{\text{ext}}) \Delta V \).