Question

Natural gas companies in the United States use the "therm" as a unit of energy. One therm is \(1 \times 10^{5} \mathrm{BTU}\). (a) How many joules are in one therm? \((1 \mathrm{~J}=\) \(\left.9.48 \times 10^{-4} \mathrm{BTU}\right)\) (b) When propane gas, \(\mathrm{C}_{3} \mathrm{H}_{8}\), is burned in oxygen, \(\mathrm{CO}_{2}\) and steam are produced. How many therms of energy are given off by \(1.00 \mathrm{~mol}\) of propane gas?

Short answer

Answer: Approximately 0.02094 therms of energy are given off when 1 mol of propane gas is combusted in oxygen.

Step-by-step solution

Convert 1 therm to Joules

Given that 1 therm is equal to \(1\times10^5\) BTUs and 1 Joule is equal to \(9.48\times10^{-4}\) BTUs, we will convert 1 therm into Joules using these conversion factors: 1 therm = \(1 \times 10^{5}\) BTUs We multiply by the given conversion of \(\mathrm{1~J}\): \(\frac{1~\mathrm{J}}{9.48 \times 10^{-4} \mathrm{BTU}}\) : 1 therm × \(\frac{1~\mathrm{J}}{9.48 \times 10^{-4} \mathrm{BTU}}\) = \(?\) Joules

Calculate the Joules in 1 therm

Now, we'll use the given conversion factors to calculate the number of Joules in 1 therm: 1 therm × \(\frac{1~\mathrm{J}}{9.48 \times 10^{-4} \mathrm{BTU}}\) = \(\frac{1 \times 10^{5}}{9.48 \times 10^{-4}}\) Joules \(\approx 1.055 \times 10^{8}\) Joules So, 1 therm is approximately \(1.055 \times 10^{8}\) Joules.

Calculate the energy released by 1 mol of propane gas

Now, we need to find out how many therms of energy are given off by 1.00 mol of propane gas (\(\mathrm{C}_{3}\mathrm{H}_{8}\)). The balanced chemical equation for the combustion of propane is: \(\mathrm{C}_{3}\mathrm{H}_{8}(g)\) + \(5\mathrm{O}_{2}(g)\) → \(3\mathrm{CO}_{2}(g)\) + \(4\mathrm{H}_{2}\mathrm{O}(g)\) The molar enthalpy of combustion of propane, \(\Delta H_{c}\), is about \(-2.21 \times 10^{6} \mathrm{J/mol}\) (we get it from a reference, such as a chemistry textbook or data table). This means that this energy is released when 1 mol of propane gas is burned: Energy released = \(-2.21 \times 10^{6} \mathrm{J}\) Note that we only take the magnitude of the enthalpy change, as we are interested in the absolute amount of energy released.

Convert the energy released to therms

To convert the energy released from Joules to therms, we'll use the conversion factor that we found in Step 1: \(\frac{1 \mathrm{J}}{1.055 \times 10^{8} \mathrm{therms}}\) We now multiply the energy released by the conversion factor: \(\frac{2.21 \times 10^{6} \mathrm{J}}{1.055 \times 10^{8} \mathrm{therms}} \approx 0.02094\) therms So, 1 mol of propane gas gives off approximately 0.02094 therms of energy when combusted in oxygen.