Thermodynamics - isochoric situation

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SUMMARY

The discussion focuses on calculating the final pressure in an isochoric situation where the volume remains constant. The user correctly identifies that with constant volume, the differential volume change (dV) equals zero, leading to the equation βdT = KdP. The correct formula for determining the change in pressure (ΔP) is established as ΔP = (β/K)(T_f - T_i), where β is expansivity, K is the isothermal bulk modulus, T_f is the final temperature, and T_i is the initial temperature. The user initially misinterprets the integration process, mistakenly concluding that the final pressure equals the initial pressure.

PREREQUISITES
  • Understanding of thermodynamic concepts such as isochoric processes.
  • Familiarity with the definitions of expansivity (β) and isothermal bulk modulus (K).
  • Basic knowledge of calculus, particularly integration techniques.
  • Ability to manipulate and solve equations involving temperature and pressure.
NEXT STEPS
  • Study the derivation and applications of the ideal gas law in isochoric processes.
  • Learn about the relationship between temperature, pressure, and volume in thermodynamics.
  • Explore advanced integration techniques relevant to thermodynamic equations.
  • Investigate real-world applications of isothermal bulk modulus in material science.
USEFUL FOR

Students and professionals in physics and engineering, particularly those specializing in thermodynamics and fluid mechanics, will benefit from this discussion.

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I'm trying to calculate the final pressure. I was given initial and final temperatures as well as initial pressure, expansitivy and isothermal bulk modulus. I was also told the volume is constant.

Since volume is constant I figured dV=0

so in the formula dV=VβdT - VKdP it reduces to:

βdT=KdP

I know that I need to solve for dP but I think I'm doing something wrong in my integral because I end up with the final pressure being the same as the initial pressure which I know is wrong. How do I solve that equation for dP?
 
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Your equation is correct. Just integrate. $$\Delta P=\frac{\beta}{K}(T_f-T_i)$$
 

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