please help me with 4.1.5, please show the all details of questions abcd,thank y

Question

please help me with 4.1.5, please show the all details of questions abcd,thank you so much!

re-estimate the final temperature in part (c) above, using the condition that 5 = const For black-body radiation we know from (2.3.2) and (2.3.3) that U = 4- VT* and PV=]U, c where a is the Stefan-Boltzmann constant given by (2.3.4) and c is the velocity of light. The purpose of this question is to show how the entropy can be calculated for a system other than a simple ideal gas. Use (4.1.7a) to define S.By applying the first law to a quasistatic process, show that the entropy can be expressed as Show that in a quasistatic adiabatic volume change VT3 = const.Given that Va and Vb are the initial and final volume, respectively, show that the heat input in a quasistatic isothermal change of volume is given by Show that, for a fixed volume, the heat capacity per unit volume defined by

Explanation / Answer

a)

We must find dQ first:

==> dQ/dT = 16 (sigma/c) V T^3

==> dQ = (16 (sigma/c) V T^3) dT

Now we can find S:

dS = dQ/T

==> S = int{((16 (sigma/c) V T^3) dT)/T}

==> S = int{(16 (sigma/c) V T^2) dT}

==> S = 16 (sigma/c) V (T^3)/3

==> S = (16 sigma)/(3 c) V T^3

b)

For a quasistatic adiabatic process, entropy is constant and (16 sigma)/(3 c) is constant too, therefore "V T^3" is constant:
S = (16 sigma)/(3 c) V T^3

==> V T^3 = (16 sigma)/(3 c S)

(16 sigma)/(3 c S) is constant, therefore

==> V T^3 = constant

c)

dQ = T dS

in a isothermal process T is constant, therefore

dQ = T (16 sigma)/(3 c) (dV) T^3

==> dQ = (16 sigma)/(3 c) T^4 (dV)

==> Q = T (16 sigma)/(3 c) T^3 (Vb - Va)

d)

dS/dT = d((16 sigma)/(3 c) V T^3)/dT

dS/dT = (16 sigma/c) V T^2

==> cV = (T/V) dS/dT

==> cV = (T/V) (16 sigma/c) V T^2

==> cV = (16 sigma/c) T^3

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