A sample of air weighing 1.18 g occupies 1.0 × 10³ cm³ when kept at 300 K and 1.0 × 10⁵ Pa. When 2.0 cal of heat is added to it at constant volume, its temperature increases by 1°C. Calculate the amount of heat needed to increase the temperature of air by 1°C at constant pressure if the mechanical equivalent of heat is 4.2 × 10⁷ erg cal⁻¹. Assume that air behaves as an ideal gas.

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Solution

Here,

m = 1.18 g = 1.18×10^-3 kg

ΔQ = 2.0×4.2 J

P = 1.0×10⁶ Pa

V = 1.0×10³ cm³ = 1.0×10^-3 m³

T = 300K

Applying eqn. of state

PV = nRT

=> n = PV/RT

=> n = 1.0×10⁵×1.0×10^-3/(8.314×300)

=> n = 0.04

ΔT = 1⁰C

(ΔH)_v = nC_vΔT

2.0×4.2 = nC_v×1

=> C_v = 8.4/n= 8.4/0.04

=> C_v = 210

Again we know

C_p – C_v =R

=> C_p = R + C_v

=> C_p = 8.3 + 210

=> C_p = 218.3

Now at constant pressure

(ΔH)_p = nC_pΔT

=> (ΔH)_p = 218.3×0.04×1 = 8.732 J

In calories

=> (ΔH)_p = 8.732/4.2 = 2.08 cal

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