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In a Gobar gas plant, gobar gas is formed by bacterial fermentation of animal refuse. It mainly contains methane and its heat of combustion is -809 \[kJmo{l^{ - 1}}\] according to the following equation: $C{H_4} + 2{O_2} \to C{O_2} + 2{H_2}O$ ; $\Delta H$= -809 kJ

How much gobar gas would have to be produced per day for a small village of 50 families, if it is assumed that each family requires 20,000 kJ of energy per day? The methane content in gobar gas is 80% by mass.

Last updated date: 13th Jun 2024
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Hint: First find out the total amount of methane that will be consumed by 50 families. Then find the amount of methane that would produce this much energy and amount of gobar gas associated with it.

Complete answer:
-According to the reaction: $C{H_4} + 2{O_2} \to C{O_2} + 2{H_2}O$ , the amount of energy evolved is 809 kJ.
-Now let us check the total amount of energy required by these families. It is given that one family requires 20,000kJ of energy and there are 50 families. So, the total amount of energy required is:
  Total energy for 50 families = Consumption of energy by 1 family × number of families
                                                    = 20,000 × 50
                                                    = 10,00,000 kJ = ${10^6}$ kJ

-As we can see in the reaction that 809 kJ of energy is released when 1 mole of methane ($C{H_4}$) is burned. So we will use the unitary method to find out the amount of methane required to release ${10^6}$ kJ of energy.
               809kJ of energy is evolved for : 1 mole of $C{H_4}$= 16 g $C{H_4}$
               1 kJ of energy will be evolved for: $\dfrac{{16}}{{809}}$ g of $C{H_4}$
   ${10^6}$ kJ of energy will be evolved by: $\dfrac{{16}}{{809}} \times {10^6}$ g of $C{H_4}$
                                                                             = 19,770 g = 19.77 kg of methane

-The question says that the methane content in gobar gas is 80% by mass.
So, let the amount of gobar gas that contains 19.77 kg of methane be ‘x’ kg. So, we can say that 80% of ‘x’ is 19.77 kg.
                                            $\dfrac{{80}}{{100}} \times x = 19.77$
                                               $x = 19.77 \times \dfrac{{100}}{{80}}$
                                                X = 24.7125 kg gobar gas

So, 24.7125 kg of gobar gas will be required by 50 families.

Note: Gobar gas is mainly composed of methane (\[C{H_4}\]), carbon dioxide ($C{O_2}$), small amounts of hydrogen sulphide (${H_2}S$) and trace amounts of nitrogen (${N_2}$), hydrogen (${H_2}$) and carbon monoxide ($CO$). This gas is highly flammable due to the presence of methane and produces blue flame on burning. Thus, it is used as a source of energy. 25 times more heat can be captured by one pound of methane than by one pound of carbon dioxide.