Show that the heat capacity at constant volume CV of a Van der Waals gas is a function of the te

Show that the heat capacity at constant volume CV of a Van...

Key Concepts

Heat Capacity at Constant Volume

Heat capacity at constant volume, denoted as C_V, is a measure of the amount of heat required to raise the temperature of a system by one degree while keeping its volume constant. It is mathematically defined as the partial derivative of the internal energy with respect to temperature at constant volume. This concept is central in thermodynamics because it tells us how the energy content of a system changes with temperature under fixed volume conditions.

Van der Waals Gas

The Van der Waals gas model introduces corrections to the ideal gas law to account for the non-ideal behavior of real gases. These corrections include finite molecular size and intermolecular attractions. While the equation of state is modified compared to an ideal gas, certain thermodynamic properties, such as the heat capacity at constant volume, can remain dependent only on temperature since the added correction terms often contribute independently of temperature to the energy.

Internal Energy Dependence on Temperature

The internal energy of a gas encapsulates both kinetic and potential energy components. For many gases, especially for models like the Van der Waals gas, the temperature-dependent part of the internal energy comes mainly from kinetic energy, while potential energy corrections (due to intermolecular attractions) are often functions of volume or constants that do not affect the derivative with respect to temperature. This leads to the scenario where the derivative of internal energy with respect to temperature (i.e., C_V) remains a function solely of temperature.

Differentiation in Thermodynamics

Using calculus, particularly partial differentiation, is essential in thermodynamic derivations. When showing that heat capacity at constant volume is a function of temperature alone for a Van der Waals gas, one differentiates the internal energy with respect to temperature while holding volume constant. This mathematical technique reveals that any volume-dependent (or intermolecular interaction) corrections that do not vary with temperature drop out, solidifying the concept that C_V depends only on temperature.

Transcript

00:01 First of all, we know that the rate change of internal energy with temperature is the heat capacity at constant volume and that is cv.

00:10 So we know that cv is equal to del u divided by del t.

00:21 That is with respect to constant volume v.

00:27 Consider this as first equation.

00:38 Differentiate equation first both sides with v keeping t constant so del cv divided by del v that in respect that in respect to temperature t then that's equal to del divided by del v multiplied by del u divided by del t with respect to constant volume v and with respect to this as a whole with respect to constant volume t so this is considered this a second equation now internal energy is a function of two variables, volume v and temperature t.

02:14 So, since we know that since u, that is internal energy, that is a function of volume v, temperature t.

02:27 So that's why, then the change in value of u will be given as, so we can write that del u divided by del t with respect to constant volume v multiplied by d t plus del u divided by del v and with respect to constant volume t this is d v.

03:08 This is the value for d .u.

03:21 So now this, consider this as equation three.

03:33 Now, since u is a perfect differential, so it satisfies euler's theorem, which can be obtained by double derivative of equation three above.

03:49 So now, since it satisfies euler theorem, so double derivative of equation 3, that is del divided by del v, multiplied by del u divided by del t, with respect to constant volume v and with respect to constant temperature t, then this is equals to del divided by del t and then we have del u divided by del v with respect to temperature t and this consider this as equation fourth okay so now second different differentials of u okay second different differentials of u internal energy with respect to temperature t and v carried out in either order and becomes equal to one another this is according to euler's theorem okay now by first law of thermo dynamics so we know that d u that's equal to dq plus d w so d u so d u that is the change in internal energy d q that is the that is the heat exchange happened in system and surroundings and d w that is the work done okay so also since we know that d s that is change in entropy okay so that is given by dq divided by d t okay not d t but t only temperature t okay so d q divided by t so this is the relation for change in entropy and similarly this can also be written as dq is equal to tds.

06:43 Okay.

06:44 So now also we know that d .w.

06:52 Work done.

06:52 So that is given by minus pdv product of pressure and volume change.

06:59 Right.

06:59 So that is called work done.

07:01 So i have value for d .w and d .s.

07:06 Dq.

07:07 Okay.

07:07 With the help of d .s relation, change in entropy relation.

07:11 Got the value for dq.

07:13 So dq and d .w value will put here in this equation.

07:19 Okay.

07:20 So therefore, now we can write that d .u.

07:30 That's equal to instead of dq, i can put tds minus.

07:37 Instead of d .w, i can put pdv.

07:40 So that is the relationship.

07:43 I can consider this as equation fifth and i can say that i got this equation sixth, okay, putting values in equation fifth.

08:10 So now i have this sixth equation here finally.

08:13 Okay, now differentiate equation 6 with respect to volume v keeping s constant.

08:57 So then del u divided by del v in respect to constant, in respect to constant temperature, d, that's equal to t multiplied by del s divided by del v divided by del v divided by temperature in respect to temperature t minus p pressure okay so consider this as equation seventh now we have also a equation we have also a relation and that is the work function relation or hell modes free energy relation and we know that work function a so that is also called hellmole's free energy so that is given by a is equal to u minus t s okay that is internal energy minus temperature multiplied by entropy okay this is the relation so now differentiate both sides equation 8 so we know that da is equal to d u minus t d s minus s d t or now this becomes d q plus d w minus t d s minus s d t so this becomes t d s minus pdv minus d s minus s d t so finally d a is equal to minus pdv minus s d t d s okay t ds to this minus t ds will be cancelled out so consider this as equation ninth okay now differentiate both sides of equation nine with respect to v keeping t as constant.

13:00 So this becomes del a divided by del v with respect to temperature t and that's become minus p...

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