Question

An ideal gas expands from 100 cm3 to 200 cm3 at a constant pressure of 2.0 × 105 Pa when 50 J of heat is supplied to it. Calculate (a) the change in internal energy of the gas (b) the number of moles in the gas if the initial temperature is 300 K (c) the molar heat capacity Cp at constant pressure and (d) the molar heat capacity Cv at constant volume.

Answer 1

The Change in internal energy, number of moles, molar heat capacity Cp and the molar heat capacity Cv at constant volume is give below.

Explanation:

Given data:

Initial volume "V1" =100 cm^3

Final volume "V2" = 200 cm^3

Pressure =2×105 Pa

Heat supplied "dQ" =50 J

• (a) Change in internal energy of gas dQ = dU+dW

Implies 50=dU+2×10^5

(200−100)×10−6

implies 50=dU+2×10

dU=30 J.

• (b) Number of moles

30 = n×32×8.3×300

[U=32 nRT for mono atomic]

n = 283×3 = 2249 = 0.008

• (c) Molar heat capacity

dU = nCvdT

Cv = dUndT=

300.008×300=12.5

Cp = Cv+R=12.5+8.3=20.8

• (d)  Molar heat capacity Cv at constant volume

Cv=12.5

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Answer 2

(a) dU = 30 J

(b) For a Gas Mono atomic is 0.008.

(c) The molar heat capacity $C_p$ at constant pressure 20.8 J/mol$^-k$.

(d) The molar heat capacity $C_v$ at constant volume $C_{v}=12.5 \mathrm{J} / \mathrm{mol}-\mathrm{K}$.

Explanation:

Initial gas volume, $V_1$ = 100 $cm^3$

Final gas l volume = $V_2$ = 200 $cm^3$

constant pressure = 2 × 105 Pa  

supplied  Heat, dQ = 50  J

(a) According to thermodynamics first law

dQ = dU + dW

$d W=P \Delta V=2 \times 10^{5} \times(200-100) \times 10^{-6}=20$

⇒ 50 = dU + 2 × 10

⇒ dU = 30 J

(b) For a Gas Mono atomic,

$U=\frac{3}{2} n R T$

$30=n \times \frac{3}{2} \times 8.3 \times 300$

$n=\frac{2}{83 \times 3}=\frac{2}{249}=0.008$

(c) the molar heat capacity $C_p$ at constant pressure

$\begin{array}{l}{d U=n C_{v} d T} \\{C_{v}=\frac{d U}{n d T}=\frac{30}{(0.008 \times 300)}=12.5} \\{C_{p}=C_{v}+R=12.5+8.3=20.8 \mathrm{J} / \mathrm{mol}-K}\end{array}$

(d) the molar heat capacity $C_v$ at constant volume.

$C_{v}=12.5 \mathrm{J} / \mathrm{mol}-\mathrm{K}$