A typical bathtub can hold 106 gallons of
water. Calculate the mass of natural gas that
would need to be burned to heat the water
for a tub of this size from 56 ◦F to 87 ◦F.
Assume that the natural gas is pure methane
(CH4) and that the products of combustion
are carbon dioxide and water (liquid).
Answer in units of g
2nd part
What volume of natural gas does this correspond to at 30◦C and 1 atm?
Answer in units of L
a.
1. Convert F to C. Convert 106 gallons H2O to L, assume density of 1.0 g/mL and convert L to grams.
2. Calculate q = heat required to heat the water as described.
q = mass H2O x specific heat H2O x (Tfinal - Tinitial).
3. Write and balance the equation for the combustion of CH4.
CH4 + 2O2 ==> CO2 + 2H2O
4. Calculate heat of combustion.
dHfrxn from step 2 = (n*dHf products) - (n*dHf reactants). Call this dHrxn
5. dHrxn in kJ/16g CH4 x # g CH4 = q
Solve for #g CH4
Check my thinking.
b.
Use PV = nRT and solve for V in L. You will need to plug in for n; remember n = grams/molar mass. You know g and molar mass so you can calculate n and obtain V from there.
To calculate the mass of natural gas required to heat the water in a tub, we need to consider the specific heat capacity and the heat equation.
First, let's calculate the energy required to heat the water in the bathtub:
q = m × Cp × ΔT
where:
- q is the heat energy required
- m is the mass of water
- Cp is the specific heat capacity of water
- ΔT is the change in temperature
Given:
- The mass of water in the bathtub is 106 gallons, which is equivalent to 106 × 3.78541 kg (since 1 gallon = 3.78541 kg)
- Cp of water is 4.184 J/g°C
- ΔT is the difference between the final and initial temperatures: 87 - 56 = 31°C
Now, let's calculate q:
q = (106 × 3.78541 kg) × (4.184 J/g°C) × 31°C
By performing the calculations, we find that q ≈ 50466.124 J.
Next, we need to calculate the amount of methane (CH4) that would produce this amount of energy upon combustion. To do this, we'll consider the heat of combustion and the molecular weight of methane.
The heat of combustion of methane is -890.36 kJ/mol. This means that when one mole of methane undergoes complete combustion, it releases 890.36 kJ of energy.
The molecular weight of methane is 16.04 g/mol (since carbon has a molar mass of 12.01 g/mol and hydrogen has a molar mass of 1.01 g/mol).
Now, let's calculate the moles of methane required:
moles = (q / heat of combustion)
moles = (50466.124 J / 890.36 kJ) × (1000 J / 1 kJ) × (1 mol CH4 / 890.36 kJ) × (16.04 g / 1 mol CH4)
By performing the calculations, we find that moles ≈ 0.1123 mol.
Finally, we can calculate the mass of methane:
mass = moles × molecular weight
mass = 0.1123 mol × 16.04 g/mol
By performing the calculations, we find that the mass of methane required to heat the water in the tub is approximately 1.8 grams.
For the second part, to calculate the volume of natural gas at 30°C and 1 atm, we need to use the Ideal Gas Law:
PV = nRT
where:
- P is the pressure (1 atm)
- V is the volume
- n is the number of moles (0.1123 mol from the previous calculation)
- R is the ideal gas constant (0.0821 L·atm/mol·K)
- T is the temperature in Kelvin (30 + 273.15 K)
Now, let's rearrange the equation to solve for V:
V = (nRT) / P
V = (0.1123 mol × 0.0821 L·atm/mol·K × (30 + 273.15) K) / 1 atm
By performing the calculations, we find that the volume of natural gas at 30°C and 1 atm is approximately 3.25 L.