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

d, conspicuous consumption is when you try to increase your wealth or social status (and in the case he's trying to create the perception of wealth considering he can't really afford it)

the answer is account d