## HEAT CHANGE UNDER CONSTANT VOLUME

The reactions are carried out under constant volume or under constant pressure

conditions. Let us now arrive at an expression useful in calculating the heat

change in a system under constant volume conditions. let us assume that the work done on the system is only pressure-volume work, whereas electrical, magnetic or other types of work are not involved.

In the application of first law of thermodynamic which can be stated in any one of the following ways:

- The energy of an isolated system remains constant.
- Energy can neither be created nor destroyed although it can be changed

from one form to another

- It is not possible to construct a perpetual motion machine which can work

endlessly without the expenditure of energy. (Such a machine is known as

perpetual motion machine of the first kind.)

U is the internal energy of a system

In other to find the internal energy of a system, the following must be involved, which

Heat and work. That what brought out the equation shown below.

dU=dq+dw

The above equation is use to find the internal energy of a system with respect to change of heat and work. dw mean change of work. We shall have a detail discussion on how to fine the internal energy in our next tutorial.

Back to our discussion

dU: change of internal energy

dq: heat change

p: pressure

dV: change of volume.

In respect to find heat change at a constant volume, we have to aply the following equation, and we called it equation 1

dU = dq – pdV……………………………. 1

Or when you make (dq) the subject of the formula, the you equation become

dq = dU + pdV …………………………..2

If the process is carried out at constant volume, then our equation become

dV – pdV = 0

Hence, dqv — dU …………………….3

Now if only finite changes in the internal energy take place equation 3 becomes,

qv = ΔU …………………………………..4

What the equation mean is that, heat absorbed by a system at constant volume is exactly equal to its internal energy change.

Let us try to correlate internal energy change with heat capacity at constant

volume assuming that there is no phase change or chemical reaction. From Equation

1 to 4

dU = CvdT= nCvdT ……………………….5

This holds good for n mol of an ideal gas. Looking for Cv which constant volume can be rewritten as,

Cv =

$\left(\frac{\partial U}{\partial T}\right)$……………………………..6

What equation 6 is telling us is that heat capacity at constant volume is equal to change in internal energy per 1 K rise in temperature at constant volume.

$\left(\frac{\partial U}{\partial T}\right)v$

*The above equation is called the partial differential of internal energy with respect to*

*temperature at constant volume. It means the value U of a gas depends on V*

*and T, but only the variation in U with respect to T is measured at constant*

*volume . Interestingly for an ideal gas, U depends, only on T but not on V.*

*For example*

In order to obtain ΔU when an ideal gas is heated from temperature T1 to T2 at

constant volume, the integrated form of Equation 5 is to be used.

i.e ΔU =

${\int}_{T1}^{T2}CvdT$=

${\int}_{T1}^{T2}\overline{C}vdT$

Hence, by knowing Cv over the temperatures T1, to T2, it is possible to obtain the

value of ΔU.

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