To solve for the temperature of the gas in the engine cylinder, we will use the Ideal Gas Law equation:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure,
- [tex]\( V \)[/tex] is the volume,
- [tex]\( n \)[/tex] is the number of moles,
- [tex]\( R \)[/tex] is the ideal gas constant,
- [tex]\( T \)[/tex] is the temperature in Kelvin.
Given values:
- Volume ([tex]\( V \)[/tex]) = 0.045 liters
- Pressure ([tex]\( P \)[/tex]) = 28 atmospheres
- Number of moles ([tex]\( n \)[/tex]) = 0.020 moles
- Ideal gas constant ([tex]\( R \)[/tex]) = 0.0821 [tex]\( \frac{L \cdot atm}{mol \cdot K} \)[/tex]
We need to solve for [tex]\( T \)[/tex]. Rearranging the Ideal Gas Law equation to solve for temperature ([tex]\( T \)[/tex]):
[tex]\[ T = \frac{PV}{nR} \][/tex]
Substitute the known values into the equation:
[tex]\[ T = \frac{(28 \text{ atm})(0.045 \text{ L})}{(0.020 \text{ mol})(0.0821 \frac{L \cdot atm}{mol \cdot K})} \][/tex]
After performing the calculations:
[tex]\[ T = 767.3568818514007 \text{ K} \][/tex]
Expressing the answer to two significant figures, the temperature of the gas is:
[tex]\[ T \approx 770 \text{ K} \][/tex]
So, the temperature of the gas is [tex]\( \boxed{770} \)[/tex] K.
