Find the work done by the gas for the given volume and pressure. Assume that the pressure is inversely proportional to the volume. (See Exam

Question

Find the work done by the gas for the given volume and pressure. Assume that the pressure is inversely proportional to the volume. (See Example 6.) A quantity of gas with an initial volume of 2 cubic feet and a pressure of 1000 pounds per square foot expands to a volume of 3 cubic feet. (Round your answer to two decimal places.)

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Dâu 1 year 2021-09-04T20:47:20+00:00 1 Answers 86 views 0

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    2021-09-04T20:49:00+00:00

    Answer:

    810.93

    Step-by-step explanation:

    Let the pressure be given by P and the volume be V.

    Since pressure is inversely proportional to volume, we can write;

    P ∝ [tex]\frac{1}{V}[/tex]

    => P = [tex]\frac{c}{V}[/tex]          ————-(i)

    Where;

    c = constant of proportionality.

    When the volume of the gas is 2 cubic feet, pressure is 1000 pounds per square foot.

    V = 2 ft³

    P = 1000lb/ft²

    Substitute these values into equation (i) as follows;

    1000 = [tex]\frac{c}{2}[/tex]

    => c = 2 x 1000

    => c = 2000 lbft

    Substituting this value of c back into equation (i) gives

    P = [tex]\frac{2000}{V}[/tex]

    This is the general equation for the relation between the pressure and the volume of the given gas.

    To calculate the work done W by the gas, we use the formula

    [tex]W = \int\limits^{V_1}_{V_0} {P} \, dV[/tex]

    Where;

    V₁ = final volume of the gas = 3ft³

    V₀ = initial volume of the gas = 2ft³

    Substitute P = [tex]\frac{2000}{V}[/tex], V₁ = 3ft³ and V₀ = 2ft³

    [tex]W = \int\limits^{3}_{2} {\frac{2000}{V} } \, dV[/tex]

    Integrate

    W = 2000ln[V]³₂

    W = 2000(In[3] – ln[2])

    W = 2000(0.405465108)

    W = 810.93016

    W = 810.93 [to 2 decimal places]

    Therefore, the work done by the gas for the given pressure and volume is 810.93

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