To determine the pressure of an ideal gas under specific conditions, we can use the Ideal Gas Law, which is given by the equation [tex]\( PV = nRT \)[/tex]. Here’s a detailed, step-by-step solution:
1. Identify the given values:
- Number of moles ([tex]\( n \)[/tex]) = 33.5 mol
- Volume ([tex]\( V \)[/tex]) = 35.5 L
- Temperature in Celsius ([tex]\( T_{C} \)[/tex]) = 85.00°C
2. Convert the temperature from Celsius to Kelvin:
The Ideal Gas Law requires the temperature in Kelvin. To convert Celsius to Kelvin, use the relationship:
[tex]\[
T_{K} = T_{C} + 273.15
\][/tex]
So,
[tex]\[
T_{K} = 85.00 + 273.15 = 358.15 \text{ K}
\][/tex]
3. Use the Ideal Gas Constant [tex]\( R \)[/tex]:
The value of the ideal gas constant [tex]\( R \)[/tex] in the correct units (L.atm/(mol.K)) is:
[tex]\[
R = 0.0821 \text{ L.atm/(mol.K)}
\][/tex]
4. Substitute all the known values into the Ideal Gas Law equation and solve for pressure ([tex]\( P \)[/tex]):
[tex]\[
PV = nRT
\][/tex]
Rearrange to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{nRT}{V}
\][/tex]
Plug in the values:
[tex]\[
P = \frac{(33.5 \text{ mol}) \times (0.0821 \text{ L.atm/(mol.K)}) \times (358.15 \text{ K})}{35.5 \text{ L}}
\][/tex]
5. Calculate the pressure:
[tex]\[
P = \frac{33.5 \times 0.0821 \times 358.15}{35.5}
\][/tex]
Simplifying this:
[tex]\[
P \approx 27.74754514084507 \text{ atm}
\][/tex]
Hence, the pressure of the gas under the given conditions is approximately [tex]\( 27.75 \)[/tex] atm.
