A vessel of volume V = 30 L contains an ideal gas at 0°C. After a portion of the gas was released out, the pressure in the vessel decreased by ΔP = 0.78 atm (without any change in... A vessel of volume V = 30 L contains an ideal gas at 0°C. After a portion of the gas was released out, the pressure in the vessel decreased by ΔP = 0.78 atm (without any change in temperature). Find the mass of the discharged gas. The density of gas under normal conditions ρ = 1.3 g/L. Read more

Steps to Solve

1. Identify the initial state parameters

The initial pressure $P_1$ can be found using the ideal gas law:

$$ P_1 = \frac{nRT}{V} $$

Given that the gas is at standard temperature (0°C or 273.15 K), we can assume we know $R$ (the universal gas constant) and $T$. For this problem, we will derive the pressure using normal conditions.

2. Calculate the decrease in pressure

The final pressure after the gas is discharged, $P_2$, will be:

$$ P_2 = P_1 - \Delta P $$

Substituting the known value of $\Delta P = 0.78$ atm:

$$ P_2 = P_1 - 0.78 $$

3. Rearrange the ideal gas law for the discharged gas

To find the mass of the discharged gas, we need to express the number of moles released using the change in pressure. Using the initial pressure to find the moles:

$$ n_1 = \frac{P_1V}{RT} $$

And for the final state (after discharge):

$$ n_2 = \frac{P_2V}{RT} $$

The change in moles, which corresponds to the mass of the gas discharged, is given by:

$$ n_{discharged} = n_1 - n_2 = \frac{(P_1 - P_2)V}{RT} $$

4. Substitute equations to find change in moles

Replace $P_2$ into the equation:

$$ n_{discharged} = \frac{(0.78)V}{RT} $$

5. Convert the moles to mass

Using the density of the gas:

$$ m = n_{discharged} \times \text{molar mass} $$

Assuming the density is given as $1.3 \text{ g/L}$, and we need to convert the volumes accordingly.

The mass of the discharged gas can then be computed as:

$$ m = \text{density} \times V_{discharged} $$

Here, $V_{discharged} = V - \text{remaining volume}$, where the remaining volume can be calculated using the final pressure.

6. Final computation of the discharged mass

Calculate using the density provided:

$$ m = \text{density} \times \Delta V $$

Substituting the known values.