Consider a mixture of air and gasoline vapor in a cylinder with a piston. The original volume is 50. cm3. If the combustion of this mixture releases 990. J of energy, to what volume will the gases expand against a constant pressure of 645 torr if all the energy of combustion is converted into work to push back the piston?
To find the volume to which the gases will expand, we need to use the ideal gas law and the conservation of energy. Here's how you can go about solving this problem step by step:
Step 1: Convert the pressure to SI units.
The given pressure is given in torr, but we need to convert it to Pascals (Pa) for consistency. 1 torr is equivalent to 133.3224 Pascals. So, 645 torr is equal to (645 * 133.3224) Pa.
Step 2: Convert the original volume to SI units.
The original volume is given as 50. cm3. However, we need to convert it to cubic meters (m3) for consistency. 1 cm3 is equal to 1 x 10^-6 m3. So, 50. cm3 is equal to (50. x 10^-6) m3.
Step 3: Calculate the final volume using the ideal gas law.
The ideal gas law is PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. In this case, we can assume the number of moles and temperature remain constant. Therefore, we have:
(P1 * V1) = (P2 * V2), where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.
We can rearrange this equation to solve for V2:
V2 = (P1 * V1) / P2
Substituting the values we have:
V2 = [(645 * 133.3224) Pa * (50. x 10^-6) m3] / P2
Step 4: Convert the final volume back to cm3.
Since the given original volume is in cm3, we'll convert the final volume from m3 back to cm3 by multiplying it by (1 x 10^6).
V2 = [(645 * 133.3224) * (50. x 10^-6) * (1 x 10^6)] cm3 / P2
Step 5: Simplify and calculate the final volume.
Plug in the given pressure value, and then evaluate the expression to find the final volume.
V2 = [(645 * 133.3224) * (50. x 10^-6) * (1 x 10^6)] cm3 / P2 = 436.653 cm3 / P2
So, the gases will expand to a volume of 436.653 cm3 against a constant pressure of 645 torr if all the energy of combustion is converted into work to push back the piston.