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Jahnavi
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I have a very basic knowledge of calculus of one variable .
In the chapter on heat and thermodynamics , ideal gas law PV =nRT is given .
Then the book says, differentiating you get
PdV +VdP = nRdT .
The book doesn't explain the differentiation step .
I think , there are two ways to differentiate the gas law PV =nRT
1) Applying product rule of differentiation when a single variable is involved :
Assuming all the three P, V, T are functions of a common variable x , I can differentiate both sides of PV = nRT by x .
d[P(x)V(x)]/dx = d[nRT]dx
Applying product rule on left side I get ,
VdP/dx+PdV/dx = nRdT/dx
Eliminating dx from the denominator from both sides I get ,
VdP+PdV = nRdT
2) Taking total derivative of both sides ,
d(PV) = d(nRT)
[∂(PV)/∂P]dP + [∂(PV)/∂V]dV = [∂(nRT)/∂T]dT
This also gives PdV +VdP = nRdT
Both approaches give same result(equation) .
Is it the product rule that is applied or is it the total derivative (involving partial differentiation ) being applied here ?
Thank you
In the chapter on heat and thermodynamics , ideal gas law PV =nRT is given .
Then the book says, differentiating you get
PdV +VdP = nRdT .
The book doesn't explain the differentiation step .
I think , there are two ways to differentiate the gas law PV =nRT
1) Applying product rule of differentiation when a single variable is involved :
Assuming all the three P, V, T are functions of a common variable x , I can differentiate both sides of PV = nRT by x .
d[P(x)V(x)]/dx = d[nRT]dx
Applying product rule on left side I get ,
VdP/dx+PdV/dx = nRdT/dx
Eliminating dx from the denominator from both sides I get ,
VdP+PdV = nRdT
2) Taking total derivative of both sides ,
d(PV) = d(nRT)
[∂(PV)/∂P]dP + [∂(PV)/∂V]dV = [∂(nRT)/∂T]dT
This also gives PdV +VdP = nRdT
Both approaches give same result(equation) .
Is it the product rule that is applied or is it the total derivative (involving partial differentiation ) being applied here ?
Thank you
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