Consider a mixture of air and gasoline vapor in a cylinder with a piston. The original volume is 30. cm3. If the combustion of this mixture releases 980. J of energy, to what volume will the gases expand against a constant pressure of 640. torr if all the energy of combustion is converted into work to push back the piston?

answer in L (liters)

Well, well, let's see here. We have a mixture of air and gasoline vapor, and a combustion that releases energy. Quite the explosive combination! Now, when this energy is converted into work, it'll push back the piston and cause the gases to expand.

We know that the original volume is 30 cm³. So, if all the energy of combustion is converted into work, we can use the good ol' ideal gas law to find the final volume.

Now, the ideal gas law is PV = nRT, but we just want to find the final volume, so we'll rearrange it to V = nRT/P, where:
- V is the final volume we're after,
- n is the number of moles of gas (which we're not given, but we'll get to that),
- R is the ideal gas constant,
- T is the temperature (which we're not given either, but stay with me),
- and P is the pressure of 640 torr.

Now, about that number of moles and temperature. We don't have them, but we can assume that the gas is behaving ideally. So, we can consider the initial and final states of the gas to be at the same temperature (let's call it T).

Since we assume the number of moles of gas doesn't change, we can actually cancel n and T out of our equation, leaving us with just V = R/P.

However, we need to be sneaky here to convert from cm³ to liters. Since 1 liter is 1000 cm³, we need to divide our result by 1000 to get the final volume in liters.

So, final volume (V) = R/P = (0.0821 L·atm/mol·K) / (640 torr) = 0.000128 L.

But remember, we had to divide by 1000 to convert from cm³ to liters, so our final answer is 0.000128 / 1000 L = 1.28 x 10^-7 L.

I hope that answered your question, my friend! Just promise not to blame me if the gases expand and things get a little... explosive. Stay safe!

To solve this problem, we can use the ideal gas law and the equation for work done by a gas. The ideal gas law is given by:

PV = nRT

Where:
- P is the pressure in torr,
- V is the volume in liters,
- n is the number of moles,
- R is the ideal gas constant (0.08206 L·atm/(mol·K)),
- T is the temperature in Kelvin.

We need to calculate the final volume (Vf) of the gases. We can use the equation for work done by a gas:

Work = PΔV

Where:
- Work is the work done by the gas (in J),
- P is the constant pressure (in torr),
- ΔV is the change in volume (in liters).

We know:
- P = 640 torr
- Work = 980 J

We can convert the work from J to L·atm by dividing by 101.325 J/L·atm:
Work = 980 J / 101.325 J/L·atm = 9.65 L·atm

Now let's calculate the change in volume (ΔV). Since the original volume (Vi) is 30 cm3, we can convert it to liters by dividing by 1000:
Vi = 30 cm3 / 1000 cm3/L = 0.03 L

Using the equation for work, we have:
Work = PΔV
9.65 L·atm = 640 torr * ΔV

Now we can solve for ΔV:
ΔV = 9.65 L·atm / 640 torr = 0.0151 L

To find the final volume (Vf), we add the change in volume (ΔV) to the original volume (Vi):
Vf = Vi + ΔV
Vf = 0.03 L + 0.0151 L = 0.0451 L

So, the volume of the gases will expand to 0.0451 L against a constant pressure of 640 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 formula for work done by a gas. Here is the step-by-step explanation:

1. Convert the initial volume to liters. We are given that the original volume is 30 cm^3. To convert to liters, divide by 1000 since there are 1000 cm^3 in 1 liter. So, the initial volume is 30/1000 = 0.03 L.

2. Use the ideal gas law to determine the initial number of moles of the gas. 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 ideal gas constant, and T is the temperature. In this case, we are not given the number of moles, so we need to solve for it.

3. Rearrange the ideal gas law equation to solve for n: n = PV / RT. Since the pressure (P), volume (V), and temperature (T) are not changing, we can substitute their values into the equation. The pressure is given as 640 torr (we will convert it to atm later), the volume (V) is 0.03 L, and the temperature (T) can be assumed to be constant at room temperature, usually around 298 K. The ideal gas constant (R) = 0.0821 L·atm/mol·K.

n = (640 torr * 0.03 L) / (0.0821 L·atm/mol·K * 298 K)
n ≈ 0.763 mol

4. Convert the pressure from torr to atm. The given pressure is 640 torr. Since there are 760 torr in 1 atm, divide 640 torr by 760 to get the pressure in atm: 640 torr / 760 torr = 0.842 atm.

5. Calculate the final volume using the work done formula. The formula for work done by a gas is:
Work = -PΔV
Where P is the pressure (in atm) and ΔV is the change in volume (in liters).

Since all the energy of combustion is converted into work to push back the piston, we can say that the work done is equal to the energy released during combustion.

Work = 980 J

Rearranging the formula, we get:
ΔV = -Work / P
ΔV = -980 J / 0.842 atm
ΔV ≈ -1162 L·atm / 0.842 atm
ΔV ≈ - 1381.5 L

The change in volume is negative because the gases are expanding.

6. Calculate the final volume. The final volume is the sum of the initial volume and the change in volume:
Final volume = Initial volume + ΔV
Final volume = 0.03 L + (-1381.5 L)
Final volume ≈ -1381.5 L

Note: It's unusual to have a negative volume, so it's possible there may be an error in the calculations. Double-check your steps or the given values to ensure accuracy.

So the final volume, under the given conditions, is approximately -1381.5 L.