A typical bathtub can hold 94 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 59 ◦F to 101 ◦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.
Convert 59 F to C.
Convert 101 F to C.
Calculate q (heat) needed to raise temperature from Tinitial to Tfinal. Do that this way.
q = mass H2O x specific heat H2O x (Tfinal-Tinitial). You will need to convert 94 gallons to liters and that to mL. The number of mL will be the grams H2O assuming the density of H2O of 1.00 g/mL.
Then CH4 + 2O2 ==> CO2 + 2H2O
How much heat do you get from the rxn? That's
dHrxn = (n*dHf products)- (n*dHf reactants).
So you dHrxn for 16 g CH4. Let's call dHrxn Y.
Then 16 g x (heat needed/Y) = ? g CH4 needed.
So my work then came out to be this, but proved to be wrong.
//Chemical Equation
CH4(g) + 202(g) = CO2(g) + 2H20(l)
//Gallons -> Liters
94 x 3.785 = 355.79
//Liters -> Mililiters
355.79 * 1000 = 355790
//Mililiters -> Grams (Density)
a = 355790g H2O
//Delta Temperature
b = 23 C
q = m x s x dT
q = (355790g)(4.184J/gC)(23C)
q = 34238383.28J
//Hess's law
891 kJ/mol
16.05 x 34238383.28J / 891 = 616752.022
Could you spot my mistake?
I would have used 23.3 for delta T but that isn't that big a deal. Makes q about 3.47E7 J.
But I didn't get close to your answer for Hess's law.
I have (2*H2O + CO2)-(CH4)
(2*-187.8)+(-393.5) - 74.81 and I have something like 700 kJ or so. Check that if you will but it isn't exactly 700 kJ. That's a significant difference. But a huge difference is in the last step you have J/kJ. You need to convert that 3.47E7 J to kJ. Hope this helps.
I picked up the dH formation for H2O2 and not H2O. Sorry about that. So other than the 23.3 C as delta T, the other big change you need to make is the conversion of J to kJ.
To calculate the mass of natural gas needed to heat the water in the bathtub, we need to know the specific heat capacity of water, the temperature change, and the energy content of natural gas.
1. First, let's calculate the energy required to heat the water. We can use the equation:
Energy = mass of water × specific heat capacity × temperature change
The specific heat capacity of water is approximately 4.184 J/g°C, and the temperature change is (101 - 59) = 42°C.
2. Next, we need to convert the energy into a relevant unit for natural gas. One standard unit for energy content is the British Thermal Unit (BTU). The energy content of natural gas is approximately 1000 BTU per cubic foot.
1 BTU = 1055.06 J
1 cubic foot of natural gas ≈ 1000 BTU
3. Now, let's calculate the energy required to heat the water in terms of BTU by converting the energy in Joules to BTU:
Energy (BTU) = Energy (J) / 1055.06
4. To determine the volume of natural gas required, we need to know the energy content of methane per volume. Since methane is the primary component of natural gas, we can use its energy content.
1 cubic foot of natural gas ≈ 1000 BTU
1 mole of methane = 16 g
1 mole of methane ≈ 22,400 BTU
Therefore, 1 g of methane ≈ 1400 BTU
5. Now, let's calculate the amount of natural gas (in grams) required to provide the calculated energy (in BTU):
Mass of natural gas (g) = Energy (BTU) / Energy content of methane (BTU/g)
Finally, convert the mass from grams to the desired unit (g).
By following these steps, you will be able to calculate the mass of natural gas needed to heat the water in the bathtub.