Why Is Heat Capacity at Constant Pressure Independent of Pressure?

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SUMMARY

The heat capacity at constant pressure (cp) is independent of pressure due to the relationship established by the ideal gas law. The equation cp = CV + (∂V/∂T) * P indicates that while a gas at high pressure may desire to expand, maintaining constant pressure limits this expansion. The work done and change in internal energy are both solely dependent on temperature change, leading to the conclusion that heat flow (Q) is proportional to temperature change (ΔT). Thus, cp remains constant regardless of pressure variations.

PREREQUISITES
  • Understanding of the ideal gas law
  • Familiarity with heat capacity concepts (cp and CV)
  • Knowledge of thermodynamic principles (work and internal energy)
  • Basic calculus for interpreting derivatives (∂V/∂T)
NEXT STEPS
  • Study the derivation of the ideal gas law and its implications
  • Explore the relationship between heat capacity and temperature changes
  • Investigate the differences between constant pressure and constant volume processes
  • Learn about thermodynamic cycles and their applications in real-world scenarios
USEFUL FOR

Students of thermodynamics, physicists, and engineers seeking to deepen their understanding of heat capacity and its behavior under varying pressure conditions.

aaaa202
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Hi, with the ideal gas law we have:

cp = CV + (∂V/∂T) * P = CV + Nk (constant P)

can someone explain why it intuitively most be so that, the heat capacity at constant pressure is independent of pressure? I mean surely a gas at high pressure wants to expand more?
 
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aaaa202 said:
Hi, with the ideal gas law we have:

cp = CV + (∂V/∂T) * P = CV + Nk (constant P)

can someone explain why it intuitively most be so that, the heat capacity at constant pressure is independent of pressure? I mean surely a gas at high pressure wants to expand more?
Perhaps you could explain why you think that heat capacity should be a function of anything - why should it not be constant?

The gas wants to expand more but if you maintain constant pressure it can only so much. How much it expands depends on the change in temperature (pressure being constant): V = nRT/P so ΔV = nRΔT/P. Since the amount of work done at constant pressure is PΔV = P(nRΔT/P) = nRΔT, the amount of work done depends only on the temperature change. The change in internal energy also depends only on the temperature change. This means that the amount of heat flow, Q = ΔU + W = ΔU + PΔV (constant pressure) is proportional to the temperature change (and vice versa ie. temperature change is proportional to heat flow). So Q = CpΔT => dQ/dT = Cp = constant

AM
 
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