Question

Suppose that 2.8 moles of an ideal diatomic gas has a temperature of 1003 K, and that each molecule has a mass 2.32

Answer 1

1) no of translational degrees of freedom - 3 ( due to each dimension in which it can move)

no of rotational degrees of freedom - 2 (as it has 2 dimensions of rotation )

no of vibrational degrees of freedom for a linear molecule - 3n - 5 where n is no of atoms in molecule concerned - i.e 3x2 - 5 = 1

2)

each degree of freedom corresponds to 1/2kt energy , where k is the boltzmann constant

so , no of degrees of freedom = 3 + 2 + 1 = 6

and 2.8 moles given

so , internal energy = n/2 x R/N(=k) x T

where n is total degrees of freedom( = no of molecules x degrees of freedom for each molecule) and N is avogadro number

so , n is 6 x 2.8 x N ( 2.8 moles correspond to 2.8 x N no of molecules)

so , internal energy =6 x 2.8/2 x 8.314 x 1003 J = 70047.11 J = 70.047 KJ

3) avg translational speed = RMS speed = sqrt(3RT/M)

R = 8.314 , T = 1003 K , M = Molar Mass = 2.32 * 10^-26 x 6.022 x 10²³ (molecular mass x no of molecules in a mole )

so , trans speed = sqrt(3x8.314x1003/2.32 * 10^-26 x 6.022 x 10²³⁾

= 179062.01 J

4) Internal energy = n/2 x R/N x T

new values n = total no of degrees of rotation = 5 x 2.8 x N (as per explanation in (2))

T = 501 K

so , change in IE = new - old = 7 x 8.314 x 501 - 70047 J = -40889.802 J