Why is the Internal Energy of a Gas nCvdT?

[vkash]

Internal of a gas is nC_vdT. why it is nC_vdT.do you know any simple proof or derivation for this.

 

[Vagn]

The heat capacity at constant volume is defined as the rate at which the heat changes with respect to temperature per mole. So for an infinitesimal change we can write.
$C_{v}=\frac{dQ}{dT}$

In an isovolumetric process no work is done so dU=δQ as per the 1st law of Thermodynamics
so we can write the equation as

$U=nC_{v}dT =n \frac{dQ}{dT}_{v}dT =n \frac{dU}{dT}_{v}dT$

 

[technician]

Cv is the molar heat capacity of a gas at constant volume and is defined as 'The heat energy required to warm 1 mole of a gas through one degree when its volume is kept constant'
Gases have 2 principal heat capacities. If the gas is kept at constant pressure then Cp is the molar heat capacity for gas at constant pressure.
When heat is supplied to a gas at constant volume no external work is done therefore all of the heat energy shows as a temperature change.
When heat is supplied to a gas at constant pressure some external work is done [P(V2 - V1)]
So for a temperature rise of 1 degree extra heat energy is required to provide the external work. This essentially means that Cp is greater than Cv and it can be shown that
Cp - Cv = R (the gas constant)

 

[vkash]

The heat capacity at constant volume is defined as the rate at which the heat changes with respect to temperature per mole. So for an infinitesimal change we can write.
$C_{v}=\frac{dQ}{dT}$

In an isovolumetric process no work is done so dU=δQ as per the 1st law of Thermodynamics
so we can write the equation as

$U=nC_{v}dT =n \frac{dQ}{dT}_{v}dT =n \frac{dU}{dT}_{v}dT$

great, but there is dent in this, that is if process is not isobaric(isovolumetric) then?

Cv is the molar heat capacity of a gas at constant volume and is defined as 'The heat energy required to warm 1 mole of a gas through one degree when its volume is kept constant'
Gases have 2 principal heat capacities. If the gas is kept at constant pressure then Cp is the molar heat capacity for gas at constant pressure.
When heat is supplied to a gas at constant volume no external work is done therefore all of the heat energy shows as a temperature change.
When heat is supplied to a gas at constant pressure some external work is done [P(V2 - V1)]
So for a temperature rise of 1 degree extra heat energy is required to provide the external work. This essentially means that Cp is greater than Cv and it can be shown that
Cp - Cv = R (the gas constant)

friend you seem to tell me that C_p-C_v=R. that is not what am i asking.

after all thanks to both guys,

I think i a just a beginner in thermodynamics. so proof of all these formula are out of my scope, I hope i will learn this formula in future.

 

[Morgoth]

1st thermodynamic law:

δQ= dU + δW.
Supposing we have the general form of U=U(T,V)
then its differential:
dU=\frac{\partial U(T,V)}{\partial T} dT + \frac{\partial U(T,V)}{\partial V} dV

and the work is δW=pdV

we go to the 1st law and replace δW and dU by the quantities we have above. We get:

δQ=\frac{\partial U(T,V)}{\partial T} dT + [\frac{\partial U(T,V)}{\partial V} + p ] dV

in case of dV=0 (V:const) you get
\frac{\partial U(T,V)}{\partial T} =δQ/dT \equiv C_v

From that you totally see that:

U= n C_v dT (for n moles now)

of course that is for Cv constant, which of course is true for ideal gases.

 

[technician]

Your original question was :
'Internal of a gas is nCvdT. why it is nCvdT.do you know any simple proof or derivation for this.'
The answer is :At constant volume no external work is done by (or on) the gas. Therefore the heat supplied = increase in internal energy.
To calculate the effect of heat supplied you nedd an 'SHC' equation
In general Heat energy = mass x SHC x temp change.
For a gas H = n x Cv x ΔT (n = number of moles rather than mass and C = molar heat capacity rather than specific heat capacity... specific means 'per kg')

If you need to know the equation for when the pressure is kept constant you need a different C... Cp. If you need something in between then you need something other than the principal Cv and Cp

 

[I like Serena]

The general equation for dU in terms of temperature and volume is:
dU=n C_V dT + n \left[ T \left({\partial P \over \partial T}\right)_V - P \right] dV
(See wikipedia)
This result can be derived from the general formula dU=TdS-PdV, which is in terms of entropy and volume.
I'll leave the proof for that out (for now).If you substitute the ideal gas law P={nRT \over V}, the requested result follows.

 

[technician]

In words:
Heat energy supplied = heat energy to raise temperature + heat energy converted to external work.(basically P x ΔV)
If there is no external work (constant vol) then
Heat energy = heat energy to raise temp