How to Calculate Temperature Change in Ideal Gas Law Problem for One Mole?

[dk_ch]

Homework Statement

for one mole of an ideal gas this relation holds good
P= P₀ /{1+(V₀/V)^2}, where P₀and V₀ are constants,
what will be its change in temperature in terms of P₀,V₀ and R,
if volume is doubled?

Homework Equations

for one mole of ideal gas we know PV=RT

Then RT/V= P₀ /{1+(V₀/V)^2}

How to proceed then? Can anyone help, please.

The Attempt at a Solution

 

[CAF123]

Evaluate the temperature of the gas when it occupies a volume V and 2V using the ideal gas law.

 

[dk_ch]

Evaluate the temperature of the gas when it occupies a volume V and 2V using the ideal gas law.

The Answer given in the text is(11/10) P₀V₀/R. How shall I reach?

 

[Chestermiller]

According to the equation, if the initial volume is Vi, what is the initial pressure Pi? What is the initial temperature (using the ideal gas law? In terms of Vi, what is the final volume, Vf? What is the final pressure?

Chet

 

[dk_ch]

According to the equation, if the initial volume is Vi, what is the initial pressure Pi? What is the initial temperature (using the ideal gas law? In terms of Vi, what is the final volume, Vf? What is the final pressure?

Chet

If we take initial pressure P₀
and initial volume as V₀
Initial temp = T₁
Final Pressure = P₀ (remaining constant)
final doubled volume =2V₀
and final temp = T₂

then by ideal gas law for one mole of gas
T₂-T₁=2P₀V₀/R -P₀V₀/R=P₀V₀/R

Is this a correct solution? Then what is the utility of the given equation?

 

[Chestermiller]

If we take initial pressure P₀
and initial volume as V₀
Initial temp = T₁
Final Pressure = P₀ (remaining constant)
final doubled volume =2V₀
and final temp = T₂

then by ideal gas law for one mole of gas
T₂-T₁=2P₀V₀/R -P₀V₀/R=P₀V₀/R

Is this a correct solution?

No. If V₀ is the initial volume, then, from the equation they gave, P₀ is not the initial pressure. As they said in the problem statement, P₀ and V₀ are just constants for the problem (and are not related to the initial conditions).

Chet

 

[dk_ch]

for initial vol V₀ the initial pressure is P₀/2
for final vol 2V₀ the final pressure is (4/5)P₀
Then temperature change becomes =(11/10)P₀V₀/R

I think it is correct

 

[Chestermiller]

for initial vol V₀ the initial pressure is P₀/2
for final vol 2V₀ the final pressure is (4/5)P₀
Then temperature change becomes =(11/10)P₀V₀/R

I think it is correct

Yes, if the initial volume is V₀.

Chet

 

[dk_ch]

Is it possible to arrive at the same result without having taken the initial volume as V₀?
If possible please suggest a way out.

 

[Chestermiller]

It might be. The problem statement sort of implies this. Have you tried, and, if so, how far have you gotten?

Chet

 

[dk_ch]

It might be. The problem statement sort of implies this. Have you tried, and, if so, how far have you gotten?

Chet

I tried but failed to eliminate v1 initial volume to have the required relation. Please give hints if possible.

 

[Chestermiller]

I'll give it a shot.

Chet

 

[Chestermiller]

The only initial volume that matches the answer in the text is it is equal to Vo.

Chet

 

[dk_ch]

So u have reached the same conclusion as I have , Thanks.