Question

Develop an expression for the higher heating value of a gaseous alkane \(C_{n} H_{2 n+2}\) in terms of \(n\).

Short answer

" To find the higher heating value (HHV) of a gaseous alkane represented as \(C_{n}H_{2 n+2}\), you need to follow these steps: 1. Write the general combustion reaction for gaseous alkanes: \(C_{n}H_{2 n+2} + \left ( n+\frac{n}{2} \right )O_{2} \rightarrow nCO_{2} + (n+1)H_{2}O\). 2. Determine the standard enthalpy (∆Hf°) of formation for the compounds in the reaction, including the given empirical expression for gaseous alkanes: \(\Delta H_{f,alkane}°=-(\ 167.5n + 19.8(2n+2) \ )\ kJ\cdot mol^{-1}\). 3. Calculate the heat of reaction (∆H°) using the standard enthalpy of formation values: \(\Delta H° = n\Delta H_{f,CO2}° + (n+1)\Delta H_{f,H2O}° - \Delta H_{f,alkane}°\). 4. Develop the expression for the higher heating value (HHV) in terms of n by simplifying the heat of reaction equation: \(HHV = -\ 240.3n - 21.2\ kJ·mol^{-1}\). By following these steps, you can calculate the higher heating value of a gaseous alkane in terms of the number of carbon atoms (n) in its molecular formula.

Step-by-step solution

Write the general combustion reaction for gaseous alkanes

To determine the higher heating value, we first need to write the balanced combustion reaction for a gaseous alkane of the form \(C_{n}H_{2 n+2}\). The reaction involves the complete combustion of the alkane with oxygen to produce carbon dioxide (CO2) and water (H2O) vapor. The general equation is given by: \(C_{n}H_{2 n+2} + \left ( n+\frac{n}{2} \right )O_{2} \rightarrow nCO_{2} + (n+1)H_{2}O\)

Determine the standard enthalpy of formation for each compound in the reaction

To determine the higher heating value, we will first determine the heat of formation (∆Hf°) for each compound in the balanced combustion reaction. For \(C_{n}H_{2 n+2}\): The heat of formation of a gaseous alkane can be determined using the following empirical expression: \(\Delta H_{f,alkane}°=-(\ 167.5n + 19.8(2n+2) \ )\ kJ\cdot mol^{-1}\) For O2: The heat of formation of elemental oxygen is 0, since it is in its standard state. For CO2: The standard enthalpy of formation for CO2 is -393.5 kJ·mol⁻¹. For H2O (gas): The standard enthalpy of formation for gaseous water is -241.8 kJ·mol⁻¹.

Calculate the heat of reaction

Now we need to calculate the heat of reaction (∆H°) using the standard enthalpy of formation values for the products and reactants in the balanced combustion reaction. The general formula for finding the heat of reaction is: \(\Delta H° = \sum(\Delta H_{f,products}°\times\ moles) - \sum(\Delta H_{f,reactants}°\times\ moles)\) \(\Delta H° = n\Delta H_{f,CO2}° + (n+1)\Delta H_{f,H2O}° - \Delta H_{f,alkane}°\) Where, - n = number of carbon atoms in the alkane. - \(\Delta H_{f,CO2}°\) = -393.5 kJ·mol⁻¹ (Standard enthalpy of formation for CO2) - \(\Delta H_{f,H2O}°\) = -241.8 kJ·mol⁻¹ (Standard enthalpy of formation for H2O)

Develop the expression for the higher heating value of an alkane in terms of n

The heat of reaction we calculated in Step 3 is equal to the higher heating value (HHV) of the gaseous alkane. We can now substitute the expression for the standard enthalpy of formation of a gaseous alkane from step 2 to develop the final expression for the higher heating value in terms of n. \(HHV = n(-\ 393.5) + (n+1)(-\ 241.8) - \left[-\ (\ 167.5n + 19.8(2n+2) \ )\right]\) By simplifying the expression, we get the final expression for HHV: \(HHV = -\ 240.3n - 21.2\ kJ·mol^{-1}\)

Key concepts

Combustion Reaction

Combustion reactions are a type of chemical reaction where a substance (typically a fuel) combines with oxygen to produce heat and light. In the context of gaseous alkanes, this process involves the complete burning of the alkane to create carbon dioxide (CO2) and water (H2O) vapor.

When we write a balanced equation for such a combustion reaction, we need to ensure that the number of atoms for each element is the same on both sides of the equation. For a given alkane with a chemical formula of \(C_{n}H_{2n+2}\), the general balanced reaction with oxygen (O2) looks like this: \(C_{n}H_{2n+2} + (n+\frac{n}{2})O_{2} \rightarrow nCO_{2} + (n+1)H_{2}O\).

The mole ratios are crucial because they allow us to later calculate the heat of reaction and ultimately the higher heating value, which is a measure of the energy that can be obtained from a substance when it is burnt.

Standard Enthalpy of Formation

The standard enthalpy of formation, \(\Delta H_{f}°\), is a thermodynamic property that indicates the amount of energy released or absorbed when one mole of a substance is formed from its elements at standard conditions. In essence, it's the change in enthalpy when one mole of a compound is formed from its elements with all substances in their standard states at 1 atm pressure.

For our gaseous alkanes, the heat of formation is given by the empirical formula: \(\Delta H_{f,alkane}°=-(167.5n + 19.8(2n+2))\ kJ\cdot mol^{-1}\). For pure elements in their standard state, like O2 gas, this value is zero. Knowing these values enables us to calculate the total energy change during a combustion reaction.

Gaseous Alkanes

Gaseous alkanes are hydrocarbons that consist purely of carbon and hydrogen atoms, with single bonds between carbon atoms. They follow a general formula of \(C_{n}H_{2n+2}\), where 'n' represents the number of carbon atoms. Examples of gaseous alkanes include methane (CH4), ethane (C2H6), and propane (C3H8).

These hydrocarbons are significant as fuels due to their combustibility and the energy they can release during combustion. Calculating the higher heating value of gaseous alkanes helps us understand how much energy can be obtained per mole when the alkane undergoes complete combustion.

Heat of Reaction

The heat of reaction, also known as the enthalpy change of a reaction, \(\Delta H°\), reflects the total heat absorbed or released during a chemical reaction. It's the difference between the enthalpy of the products and the enthalpy of the reactants.

To find the heat of reaction for an alkane's combustion, we can use the formula: \(\Delta H° = \sum(\Delta H_{f,products}°\times moles) - \sum(\Delta H_{f,reactants}°\times moles)\). For the complete combustion of an alkane, our equation becomes \(\Delta H° = n\Delta H_{f,CO2}° + (n+1)\Delta H_{f,H2O}° - \Delta H_{f,alkane}°\), where the standard enthalpy of formation for CO2 and H2O gas are known constants. This heat of reaction is essential because it is directly related to the higher heating value of the fuel, which indicates the maximum energy available from its combustion.