To determine the Celsius temperature at which 0.750 moles of an ideal gas occupies a volume of 35.9 liters under a pressure of 114 kilopascals, we can use the Ideal Gas Law, which is given by the equation:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure in pascals (Pa)
- [tex]\( V \)[/tex] is the volume in liters (L)
- [tex]\( n \)[/tex] is the number of moles
- [tex]\( R \)[/tex] is the gas constant, which is [tex]\( 8.314 \)[/tex] J/(mol·K)
- [tex]\( T \)[/tex] is the temperature in Kelvin (K)
Given values:
- Pressure [tex]\( P = 114 \)[/tex] kPa (which should be converted to Pa)
- Volume [tex]\( V = 35.9 \)[/tex] L
- Number of moles [tex]\( n = 0.750 \)[/tex] mol
First, convert the pressure from kilopascals to pascals because [tex]\( 1 \)[/tex] kPa [tex]\( = 1000 \)[/tex] Pa:
[tex]\[ P = 114 \times 1000 = 114000 \text{ Pa} \][/tex]
Next, use the Ideal Gas Law to solve for the temperature [tex]\( T \)[/tex] in Kelvin:
[tex]\[ T = \frac{PV}{nR} \][/tex]
Substitute the known values into the equation:
[tex]\[ T = \frac{(114000 \text{ Pa}) \times (35.9 \text{ L})}{(0.750 \text{ mol}) \times (8.314 \text{ J/(mol·K)})} \][/tex]
Calculate the temperature [tex]\( T \)[/tex] in Kelvin:
[tex]\[ T \approx 656338.70579745 \text{ K} \][/tex]
Finally, convert the temperature from Kelvin to Celsius using the formula:
[tex]\[ T(°C) = T(K) - 273.15 \][/tex]
[tex]\[ T(°C) \approx 656338.70579745 - 273.15 = 656065.556 \text{ °C} \][/tex]
Rounded to three significant figures, the temperature in Celsius is:
[tex]\[ 656065 \][/tex]
