Boundary Work

The Essentials

Work is a force acting over a distance. If the boundary of a system changes location, it is creating work. This specific type of work removed from a system is known as boundary work and represents an exchange of energy between the system and the surroundings. Generally, boundary work is represented by the integral of pressure with respect to the volume of the system and thus is path function. W b = ∫ P d V W_b=\int PdV Because boundary work commonly occurs under specific conditions, a set of equations are commonly used for individual conditions. The most common of these conditions is the Isobaric condition, where there is no change in pressure. W b = P ( V 2 − V 1 ) W_b=P(V_2-V_1) Another common condition is Isothermal conditions, where there is no temperature change. W b = P 1 V 1 * ln ⁡ ( V 2 V 1 ) W_b=P_1V_1*\ln(\frac{V_2}{V_1}) Another condition is polytropic expansion, where the temperature changes with respect to the exponential coefficient n n The exponential coefficient is always less than 1. W b = P 2 V 2 − P 1 V 1 1 − n = m R ( T 2 − T 1 ) 1 − n W_b=\frac{P_2V_2-P_1V_1}{1-n}=\frac{mR(T_2-T_1)}{1-n} Boundary work is considered to be out of the system when positive and into the system when negative.

Example

1: A 0.5 m 3 ^3 ballon at 100 kPa and 20 deg C is expanded using a variety of expansion conditions. Attached to the outside of the ballon is a piston-cylinder that does work as the ballon expands. Determine the amount of work produces if the ballon expands a) with constant pressure to a volume of 2 m 3 ^3 , b) if the temperature is kept constant until the volume reaches 2 m 3 ^3 , and c) if the ballon is expanded poly-tropically with an exponential constant of 0.8 until the temperature is 100 deg C.

To solve this problem, apply the boundary work equations. The first scenario is isobaric expansion, because the pressure in the balloon is maintained constant. Apply the given information the isobaric boundary work equation. W b = P ( V 2 − V 1 ) = 100 ( 2 − 0.5 ) = 150 kJ W_b=P(V_2-V_1)=100(2-0.5)=150\text{ kJ} Part b describes isothermal expansion, where the temperature is constant for the extent of the expansion. Apply the isothermal boundary work equation. W b = P 1 V 1 * ln ⁡ ( V 2 V 1 ) = 100 ( 0.5 ) * ln ⁡ ( 2 0.5 ) = 69.31 kJ W_b=P_1V_1*\ln (\frac{V_2}{V_1})=100(0.5)*\ln (\frac{2}{0.5})=69.31\text{ kJ} Part C describes polytropic expansion. Apply the polytropic boundary work equation. To do this, first use the ideal gas equation to solve for the mass of the air in the system. P V = m R T → m = P V R T = 100 ( 0.5 ) 0.287 ( 273 + 20 ) = 0.5945 kg PV=mRT\rightarrow m=\frac{PV}{RT}=\frac{100(0.5)}{0.287(273+20)}=0.5945\text{ kg} Then use the ideal gas form of the boundary work equation. W b = 0.5945 ( .287 ) ( 100 − 20 ) 1 − 0.8 = 68.259 kJ W_b=\frac{0.5945(.287)(100-20)}{1-0.8}=68.259\text{ kJ}

Practice

1: A piston-cylinder system at 65kPa and 27 deg C holds 0.2 kg of air. Heat is added to the system until the temperature is 100 deg C. How much work is extracted from the system?

2: An ideal gas is expanded while extracting work. Which type of expansion, isobaric or polytropic expansion, will result in the most work extracted, if the starting and final temperatures are the same for both processes? By what factor is the boundary work greater?

Solutions:

1: 4.1902 kJ

2: Isobaric, by a factor of ( 1 − n ) (1-n)