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Heat and First Law of Thermodynamics: Prove that Cp – Cv = R

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Physics
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Heat and First Law of
Thermodynamics:
Prove that C
p
– C
v
= R
Consider n moles of ideal gas enclosed in a cylinder having frictionless and
non-conducting piston. If the piston is #xed and the gas is heated, its volume
remains constant. In this case all the heat supplied to the system increases
the it’s internal energy due to which temperature of the gas increases.
If Q
v
is the amount of heat supplied to the gas at constant volume and ∆T is
the rise of temperature, then
C
v
=
Q
v
/n∆T
Q
v
= nC
v
∆T …….1
Since this het increases the internal energy of the gas, so we can write as
Q
v
= ∆E
int
………..2
Comparing equation (1) and (2), we have
∆E
int
= nC
v
∆T………..(3)
If the gas is heated again through the same rise of temperature ∆T at
constant pressure, then in this case, the heat supplied is used in two ways.
(1)A part of heat is used in doing work to move the piston up against the
external pressure.
(2)The other part of heat increases the internal energy of the gas.
Thus more heat is required in this case. If Q
p
is the amount of heat supplied
to the gas at constant volume and ∆T is the rise of temperature, then

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Heat and First Law of Thermodynamics: Prove that Cp - Cv = R Consider n moles of ideal gas enclosed in a cylinder having frictionless and non-conducting piston. If the piston is fixed and the gas is heated, its volume remains constant. In this case all the heat supplied to the system increases the it's internal energy due to which temperature of the gas increases. If Qv is the amount of heat supplied to the gas at constant volume and ?T is the rise of temperature, then Cv = Qv/n?T Qv = nCv ?T ??.1 Since this het increases the internal energy of the gas, so we can write as Qv = ?Eint???..2 Com ...
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