A gas company in Massachusetts charges $1.30 for 15 ft3 of natural gas (CH4) measured at 20°C and 1.0 atm. Calculate the cost of heating 200 mL of water (enough to make a cup of coffee or tea) from 20°C to 100°C. Assume that only 50 percent of the heat generated by the combustion is used to heat the water; the rest of the heat is lost to the surroundings.

$0.0224

Explanation:

First, determine the amount of heat required to raise the temperature of 450 mL water from 20°C to 100°C.

density of water = 1 g/mL  

Use = AH = m(AT)Cp

AH (delta H) = amount of heat required

m = mass of water = 450 mL: (1g/mL) = 450 g

AT(delta T) = temperature difference = 100 - 20 = 80 °C

Cp = specific heat of water (from a table) = 4.18 J/g °C

So,  

AH  = 450 (80) (4.18) = 150 480 J = 150.48 kJ

Second, determine the heat ACTUALLY required

Since 50 % of heat produced (from combustion of methane) are used for water,

heat ACTUALLY required = 2 x 150.48 kJ = 300.96 kJ

Third, determine the amount of methane in the supply (in mole)

Assume ideal gas,

PV = nRT

P = 1.0 atm

V = 17 ft3 = 481.38 L

T = 20 + 273 = 293 K  

R = 0.082 1 L atm K−1 mol−1

n = PV/RT

so,

n = (1 x 481.38)/(293 x  0.0821) = 20.011 mol

Interpretation: 17 ft3 of methane supplied is equivalent to 20.011 mol methane (17 ft3/20.011 mol)

Fourth, determine the heat produced from combustion of methane

The combustion of methane gas is represented by the reaction:

CH4 (g)  +   2 O2 (g) CO2 (g)  +   2 H2O (g)  

heat of combustion =

[heat of formation of CO2 (g) + 2 x heat of formation of H2O (g)] -

[heat of formation of CH4 (g) + 2 x heat of formation of O2 (g)]

From the standard table:

= [ -393.5 kJ + 2x(-241.8 kJ)] - [-74.8 kJ + 2x(0 kJ)]

= - 802.3 kJ  

Interpretation: The combustion of 1 mole of methane produces 802.3 kJ, otherwise known as molar enthalpy (-802.3 kJ/mol)

Fifth, determine the amount of methane required from energy requirement for heating water

heat required = 300.96 kJ

So,  

methane required = 300.96 kJ (1mol/ 802.3 kJ) =  0.3751 mol

Sixth, determine the volume of methane required

Volume methane required = 0.3751 mol (17 ft3/20.011 mol)

= 0.3186 ft3

Finally, the cost required at price rate $1.20/17 ft3

Cost of methane =  0.3186 ft3 ($1.20/17 ft3) =  

$0.0224

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