- #1
McAfee
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Homework Statement
Homework Equations
The Attempt at a Solution
I understand what the question is asking. Both ways I should get the same answer. I'm having trouble figuring out the mathematics behind this question.
Goddar said:Hi.
You're dealing with an ideal gas: what's the equation of state? (the one relating P, V and T)
Goddar said:That was your only missing piece, now you can work the integrals.
Goddar said:Use the equation of state to determine the derivatives. Example:
PV = nRT
so that, if n is constant:
VΔP + PΔV = nRΔT
When P is constant: [itex]\frac{\partial V}{\partial T}[/itex] = [itex]\frac{nR}{P}[/itex],
When T is constant: [itex]\frac{\partial V}{\partial P}[/itex] = – [itex]\frac{nRT}{P^{2}}[/itex]
Just pay attention to what's constant and what's not when following your equations. Take the time to understand what you're doing, but you have everything you need.
Goddar said:Ok, you have to be careful when integrating. First with:
ΔV1-2-3 = ([itex]\frac{\partial V}{\partial T}[/itex])P1 ΔT + ([itex]\frac{\partial V}{\partial P}[/itex])T2 ΔP,
You get:
∫ΔV1-2-3 = Vfinal – Vinitial = ∫([itex]\frac{\partial V}{\partial T}[/itex])P1 ΔT +∫([itex]\frac{\partial V}{\partial P}[/itex])T2 ΔP = [V(P1,T2) – V(P1,T1)] + [V(T2,P2) – V(T2,P1)]
= R([itex]\frac{ΔT}{P_{1}}[/itex] + [itex]\frac{T_{2}}{P_{2}}[/itex] – [itex]\frac{T_{2}}{P_{1}}[/itex])
So you see that using integrations is purely formal here: at the end you don't have to differentiate or integrate anything.
Also, be careful with your units: atm is not a S.I. unit, so the gas constant must be adjusted if you use atms...
"Show that the Change in Volume is Independent of the Path" is a statement in thermodynamics which means that the change in volume of a system is solely dependent on the initial and final states of the system, and not on the path taken to reach those states.
It is important to show that the Change in Volume is Independent of the Path because it is a fundamental principle in thermodynamics that helps us understand and predict the behavior of gases and liquids. It also allows us to simplify calculations and make accurate predictions about the behavior of a system.
The Change in Volume is independent of the path taken because it is determined by the initial and final states of a system, and not by the specific steps or processes used to reach those states. This means that as long as the initial and final states are the same, the change in volume will be the same, regardless of the path taken.
One example is the expansion of a gas in a container. If the initial and final states are the same, the change in volume will be the same regardless of whether the gas expands slowly or rapidly, or if it undergoes a series of intermediate steps. The only important factor is the initial and final states of the gas.
The independence of the Change in Volume is related to other fundamental thermodynamic principles, such as the first law of thermodynamics (conservation of energy) and the second law of thermodynamics (entropy). It also relates to the ideal gas law, which states that the change in volume of an ideal gas is directly proportional to the change in temperature, and inversely proportional to the pressure and amount of gas.