A monatomic gas fills the left end of the cylinder in the following figure. At 3

Question

A monatomic gas fills the left end of the cylinder in the following figure. At 300 K , the gas cylinder length is 13.0 cm and the spring is compressed by 5.0 cm .(Figure 1)

Problem 17.82 Part A mic gas fills the left end of the cylinder in the A monatomic gas fills the left end of the cylinder in the following fiqure. At 300 K, the gas cylinder length is 13.0 cm and the spring is compressed by 5.0 cm (Figure 1) How much heat energy must be added to the gas to expand the cylinder length to 16.0 cm? Express your answer with the appropriate units. alueUnits Figure 1 of 1 Submit My Answers Give Up Vacuum 8.0 cm2 2000 N/m

Explanation / Answer

The ideal gas equation,

pV=nRT,

p_i = (2000 N/m × 0.05 m) / 0.0008 m2 = 12500 Pa

V_i = 0.0008 m2 × 0.13 m = 1.04×10-4 m3

Numbe of moles, n = (12500 Pa × 1.04×10-4 m3) / (8.31 J/molK × 300 K) = 5.21×10-3 mol

Number of moles is constant,

p_f = (2000 N/m × 0.08 m) / 0.0008 m2 = 2×105 Pa

V_f = 0.0008 m2 × 0.16 m = 1.28×10-4 m3

T_f = (2×105 Pa × 1.28×10-4 m3) / (5.21×10-3 × 8.31 J/molK) = 591 K

so T = 291 K

Considering the 1st Law aspect of the problem,

change in thermal energy = change in temperature of any ideal-gas process:

i.e. U = nCT

where, C is the isochoric molar specific heat which is equal to 3R/2 in this case for monoatomic.

U = 5.21*10^-3*12.5*291 = 19.0 J

since Q = U - W

where W = 0.5k(xi^2-xf^2) and xi = 0.05 m, xf = 0.08 m

so W = 0.5*2000*[0.05^2 - 0.08^2 ] = -3.90 J

Q = 19.0 J - (-3.90 J) = 22.9 J

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