A steam engine goes through the following 3 step process.
I) Isobaric compression from a volume of 100 L to 10 L at a
pressure of 4 x 105 Pa
II) Isovolumetric process from a pressure of 4 x 105 Pa to a
pressure of 4 x 106 Pa at a volume of 10L.
III) An isothermal expansion from 4 x 106 Pa and a volume of 10 L
to a pressure of 4 x 105 Pa and volume of 100L, at 208 ºC
a) Using the end of step III, determine the number of moles of steam.
b) How much work is done on the gas by each step?
c) At the end of the cycle, how much heat has been added or removed from the engine?
a) Use the ideal gas law. You know P, V, and T before the expansion
n = PV/RT
n is the number of moles.
b) Use the definition of Work
Win = -Wout = - Integral of P dV.
For first step, III, P is constant so
Work in = -P*(change of V)= 4*10^5 * 90
Try doing the other steps yourself.
c)Use
Qout = Qin + Win
(Conservation of energy for a single cycle)
To determine the number of moles of steam at the end of step III, we can use the ideal gas law. The ideal gas law states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
In this case, we know the pressure (4 x 105 Pa), volume (100 L), and temperature (208 ºC = 208 + 273 = 481 K) at the end of step III. We can rearrange the ideal gas law equation to solve for n:
n = PV / RT
Plugging in the values, we get:
n = (4 x 105 Pa) * (100 L) / (8.314 J/(mol·K) * 481 K)
Simplifying the expression gives us the number of moles of steam.
For the second part, to determine the work done on the gas by each step, we can use the definition of work. In this case, work is equal to the negative of the integral of pressure (P) with respect to volume (V).
In the first step, III, the pressure is constant, so the work done can be calculated by multiplying the pressure by the change in volume:
Work in = -P * (change in V) = - (4 x 105 Pa) * (10 L - 100 L) = -4 x 105 Pa * (-90 L)
For the other steps, you can follow the same approach, considering whether the pressure or volume is constant and calculating the corresponding work done.
Finally, to determine the amount of heat added or removed from the engine at the end of the cycle, we can use the conservation of energy. The heat added (Qin) is equal to the heat output (Qout) plus the work done (Win):
Qout = Qin + Win
Since the work done is already calculated for each step, you can substitute the values and calculate the heat added or removed from the engine.