\(1.00\) litre sample of a mixture of \(\mathrm{CH}_{4(\mathrm{~g})}\) and \(\mathrm{O}_{2(\mathrm{~g})}\) measured at \(25^{\circ} \mathrm{C}\) and 740 torr was allowed to react at constant pressure in a calorimeter which together with its contents had a heat capacity of \(1260 \mathrm{cal} / \mathrm{K}\). The complete combustion of the methane to \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) caused a temperature rise in the calorimeter of \(0.667 \mathrm{~K}\). What was the mole per cent of \(\mathrm{CH}_{4}\) in the original mixture? \(\Delta H_{\text {comb }}^{\circ}\left(\mathrm{CH}_{4}\right)=-215 \mathrm{k}\) cal \(\mathrm{mol}^{-1}\).

The Ideal Gas Law is a fundamental principle in chemistry that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas. Expressed by the equation PV = nRT, where R is the gas constant, this law allows us to calculate any one of these variables if the others are known.

For the problem at hand, we used the Ideal Gas Law to determine the amount of methane in moles. Knowing the volume of the sample, the temperature in Kelvin, and the pressure in atmospheres, we could solve for n, the number of moles of methane present before combustion. It's important to convert all measurements to their standard units: liters for volume, Kelvin for temperature, and atmospheres for pressure, to ensure the correct application of the gas constant, R.

We must also remember that real gases may not behave exactly like ideal gases, but the Ideal Gas Law provides a good approximation for many conditions, particularly those close to standard temperature and pressure.

Calorimetry is the measurement of heat flow in a chemical reaction, physical change, or heat capacity. In our exercise, we are using calorimetry to find out the amount of heat produced by the combustion of methane.The calorimeter had a known heat capacity, and by observing the change in temperature, we could calculate the heat released using the formula q = mcΔT. Because we were given a constant pressure calorimeter, the heat capacity used in the calculation was at constant pressure. The value CΔT gave us the total heat q released by the reaction. This step is critical to understanding how much energy is produced by the combustion of a known amount of methane. Understanding calorimetry is essential in fields that examine energy changes in chemical processes, such as thermodynamics.

Dalton's Law of Partial Pressures is a gas law that states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. This law implies that gases in a mixture each exert pressure independently of each other.In this exercise, we assumed that all the pressure was due to methane before the reaction. Since methane and oxygen were the only gases present and they react completely, the initial partial pressure of methane can be determined from the total pressure. This is an essential step because it sets the stage for using the Ideal Gas Law to calculate the moles of methane in the reaction. Dalton's Law is an important concept in calculating the composition of gas mixtures and understanding the behavior of gases in various conditions.

Enthalpy of combustion is a chemical property that signifies the heat released when one mole of a substance burns completely in the presence of oxygen. It is usually expressed in energy units per mole, such as kilocalories per mole (kcal/mol). The negative sign indicates the release of heat, making the process exothermic.In the problem provided, the combustion enthalpy of methane is given as -215 kcal/mol. This value helped us determine the amount of heat released during the reaction. By dividing the total heat released by this enthalpy value, we can find the number of moles of methane that combusted. This step is instrumental to finding the mole percent of methane in the original mixture, which ties back into our understanding of calorimetry and the Ideal Gas Law. Enthalpy changes, such as the enthalpy of combustion, are vital for quantifying energy changes in reactions and for gauging the efficiency of fuel sources.