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 955 J of energy, to what volume will the gases expand against a constant pressure of 675 torr if all the energy of combustion is converted into work to push back the piston?
To solve this problem, we need to use the ideal gas law and the relationship between work, pressure, and volume.
1. Convert the original volume to liters: 50 cm3 = 50 / 1000 = 0.05 L
2. Convert the pressure to atm: 675 torr = 675/760 = 0.8882 atm
3. Write down the ideal gas law equation: PV = nRT
4. Rearrange the equation to solve for the number of moles: n = PV / RT
5. Since it's a mixture of air and gasoline vapor, the value of the gas constant (R) will depend on the average molar mass of the mixture. For simplicity, we can use the average molar mass of gasoline vapor, which is approximately 58.1 g/mol.
6. Convert the molar mass to kg/mol: 58.1 g/mol = 0.0581 kg/mol
7. Convert the energy release to joules: 955 J
8. The work done (W) by the gases expanding against a constant pressure is given by: W = -PΔV, where ΔV is the change in volume.
9. Rearrange the equation to solve for ΔV: ΔV = -W / P
10. Substitute the given values into the equation: ΔV = -955 J / (0.8882 atm * 0.05 L)
11. Convert the volume change to liters: ΔV = -955 J / (0.8882 atm * 0.05 L) = -2157.79
12. The negative sign indicates that the volume decreases. Since volume cannot be negative, we take the magnitude of ΔV: ΔV = 2157.79 L
13. Finally, calculate the final volume: Final volume = Original volume + ΔV = 0.05 L + 2157.79 L = 2157.84 L
Therefore, the final volume to which the gases will expand against a constant pressure of 675 torr is approximately 2157.84 liters.
To determine the final volume of the gases, we can use the ideal gas law equation, PV = nRT, where P represents pressure, V represents volume, n represents the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
In this case, we know the initial volume (V1) is 50 cm³ and the initial pressure (P1) is constant at 675 torr. We need to calculate the initial number of moles (n1).
To find the number of moles, we can use the equation n = PV / RT, where P is the pressure, V is the volume, R is the ideal gas constant (0.0821 L·atm/(mol·K)), and T is the temperature in Kelvin.
Since the temperature is not given in the problem, we can assume it remains constant throughout the process. Therefore, we can use the same temperature for the initial and final states, denoted as T1.
Next, we need to calculate the initial number of moles (n1) using the given pressure and volume.
n1 = (P1 * V1) / (R * T1)
Now that we have determined the initial number of moles of gas, we can proceed to find the final volume (V2) using the concept of energy conversion. The energy released during combustion, in this case, is converted into work to push back the piston. Therefore, the energy released is equal to the work done.
The work done by a gas is defined as the area under the curve on a pressure-volume (P-V) diagram. In this case, since the pressure is constant, the work done can be calculated using W = PΔV, where W is work, P is pressure, and ΔV represents the change in volume.
We can rearrange this equation to solve for ΔV:
ΔV = W / P
Since we want to find the final volume (V2), we need to calculate the change in volume (ΔV) using the given energy (W) and the constant pressure (P1).
ΔV = W / P1
Finally, we can find the final volume (V2) by adding the change in volume (ΔV) to the initial volume (V1).
V2 = V1 + ΔV
Substituting the previously calculated values, we can find the final volume (V2).