Specific heat capacity

Monatomic gas: Using the law of equiportition of energy the average energy of molecule at temperature.

                     

Total internal energy of a gas is 

C_v [Monatomic gas] = 

For an ideal gas.

C_P - C_V = R

C_P = J/2 R

Ratio of specific heats.

Diatomic gases: Total internal energy [law of equiportition of energy].

Polyatomic gases: We can write the total internal energy 

i.e., 

C_p = (4 + f) R.

Where f = no. of vibrational modes 

Specific heat capacity of solids

1. Consider a solid of “N” atoms, each are vibrating about its mean position.

2. Average energy = 2 x ½ K_BT [one - dimension]

3. Average in three dimension = 3K_BT

4. u = n x 3K_BT x N_A   At solids DY is negligible

So DQ = DU.

Specific heat capacity of heater:

u = n x 3K_BT x N_A = 3nRT

n = Number of atoms in molecules

c =  = 3nR

Example: The average KE and the rms velocity of the molecule in a sample of oxygen gas at 300k are 6.21 x 10⁻²¹ and 484m/s respectively. If the temperature is doubled find the value new KE and rms speed.

a) KE = 12.42 x 10⁻²²rms = 685 m/s

b) KE = 11.42 x 10⁻²²rms = 686.5 m/s

c) KE = 11.42 x 10⁻²³rms = 684.4 m/s

d) None of these

Answer d)

Solution: rms, speed .

On doubling temperature KE = 2E and rms speed = √2c.