Question

A ten-gallon methane tank contains \(1.243 \mathrm{~mol}\) of methane \(\left(\mathrm{CH}_{4}\right)\) at \(74^{\circ} \mathrm{F}\). Express the volume of the tank in liters, the amount of methane in the tank in grams, and the temperature of the tank in Kelvin.

Short answer

Answer: The volume of the tank is 37.8541 liters, the amount of methane in the tank is 19.94415 grams, and the temperature of the tank is 296.48 Kelvin.

Step-by-step solution

Convert the volume from gallons to liters

To convert the volume of the 10-gallon methane tank to liters, use the conversion factor 1 gallon = 3.78541 liters.#tag_latex#\text{Volume in liters} = 10\,\text{gallons} \times \frac {3.78541\,\text{liters}} {1\,\text{gallon}} = 37.8541\,\text{liters}#

Convert the amount of methane from moles to grams

The molecular weight of methane (CH4) is 12.01 g/mol for carbon and 1.01 g/mol for each hydrogen. Thus, the total molecular weight of methane is: #tag_latex#(\text{Molar mass of } \mathrm{CH}_{4}) = 12.01\,\text{g/mol} + 4 \times 1.01\,\text{g/mol} = 16.05\,\text{g/mol}# Next, multiply the given moles of methane by the molar mass of methane to convert it to grams.#tag_latex#\text{Mass of methane in grams} = 1.243\,\text{mol} \times 16.05\,\frac{\text{g}}{\text{mol}} = 19.94415\,\text{g}#

Convert the temperature from Fahrenheit to Kelvin

To convert the temperature from Fahrenheit to Kelvin, first, we need to convert it to Celsius. The conversion formula is: #tag_latex#T_{\text{Celsius}} = \frac{5}{9} \times(T_{\text{Fahrenheit}} - 32)# Next, we'll convert the Celsius temperature to Kelvin using the formula: #tag_latex#T_{\text{Kelvin}} = T_{\text{Celsius}} + 273.15# Calculate the temperature in Celsius:#tag_latex#T_{\text{Celsius}} = \frac{5}{9} \times(74^{\circ}\mathrm{F} - 32)=23.33^{\circ}\mathrm{C}# Now, convert Celsius to Kelvin:#tag_latex#T_{\text{Kelvin}} = 23.33^{\circ}\mathrm{C} + 273.15 = 296.48\,\mathrm{K}# #Summary# The volume of the tank is 37.8541 liters, the amount of methane in the tank is 19.94415 grams, and the temperature of the tank is 296.48 Kelvin.