Kinetic Energy of a gas Ktg

Kinetic Energy of a gas molecule

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The Pressure of an ideal gas is given by ,

P= 1/3 * M/V  * C²_rms --------(1)

P= 1/3 * Nm/V  * C²_{rms --------} ( M= Nm)

PV = 1/3 * Nm * C²_rms

PV=2/3 * N * (1/2 *mC²_rms)

The term (1/2 *mC²_rms) in above equation give kinetic energy of  a gas molecule. 

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Show that (C²_rms) ∝ ✓T

From equation (1) above

P= 1/3 * M/V  * C²_rms 

Let consider 1 mole of gas in a container having volume V .Let P be the presure and T  be the temperature

PV= 1/3 * M  * C²_rms 

But we know that PV = RT when n=1

PV= 1/3 * M  * C²_rms 

3RT/M =  C²_rms 

Hence from above equation we can say that rms velocity of a molecule of gas is directly proportional  to square root of its absolute temperature.

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Kinetic Energy of a gas in terms of pressure and volume

Kinetic energy of a gas is given by ,

P= 1/3 * M/V  * C²_rms 

3PV= M  * C²_rms 

Dividing equation by 2

3/2 PV = 1/2 * MC²_rms 

We know that 1/2 * MC²_rms  is kinetic energy

3/2 PV = KE

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Kinetic Energy Per unit volume

KE =  3/2 PV

KE per unit Volume = (Kinetic energy of a gas / Volume of  a gas)

= [(3/2 PV ) / V]

KE / Volume = 3/2 P

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Kinetic Energy Per mole

KE =  3/2 PV

 Ideal gas equation is given by,

PV = nRT

KE =  3/2 nRT

 for one mole,

 

KE =  3/2 RT

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Kinetic Energy Per molecule

KE per mole  =  3/2 RT

Avagadros Number = N_A 

KE / Molecule = 3/2 * RT  / N_A 

But R/ N_A= K_b

KE / Molecule =  3/2 K_bT 

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Kinetic Energy Per unit mass

KE / Mass = (3/2 RT) / M

KE / Mass = 3RT / 2M

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