1. A typical bathtub can hold 102 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 63 ◦F to 98 ◦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.
2. What volume of natural gas does this correspond to at 35◦C and 2 atm?
Answer in units of L.
Can you please explain to me how I should do these problems?
Sure, I'd be happy to help you with these problems!
1. To calculate the mass of natural gas needed to heat the water, we need to use the heat capacity and the heat of combustion of methane. The specific heat capacity of water is 4.18 J/g°C, and the heat of combustion of methane is approximately 891 kJ/mol.
First, let's calculate the heat required to heat the water. The temperature change is given by:
ΔT = final temperature - initial temperature = 98 - 63 = 35°C
The heat required can be calculated using the formula:
q = m * c * ΔT
Where:
q is the heat required
m is the mass of water
c is the specific heat capacity of water
ΔT is the temperature change
Since we know the specific heat capacity of water and the temperature change, we can rearrange the formula to find the mass:
m = q / (c * ΔT)
m = (102 gallons * 3785 g/gallon) / (4.18 J/g°C * 35°C)
m = 39,466.867 g
Now, let's find the moles of methane required. The molar mass of methane (CH4) is approximately 16 g/mol (12 g/mol for carbon + 4 g/mol for hydrogen).
moles = mass / molar mass
moles = 39,466.867 g / 16 g/mol
moles = 2466.679 mol
Finally, let's find the mass of methane required. Since the balanced equation for the combustion of methane is:
CH4 + 2O2 → CO2 + 2H2O
The stoichiometry is 1 mol methane to 1 mol carbon dioxide. The molar mass of carbon dioxide is approximately 44 g/mol.
mass of methane = moles of methane * molar mass of methane
mass of methane = 2466.679 mol * 16 g/mol
mass of methane = 39,466.867 g
Therefore, the mass of natural gas needed to heat the water is approximately 39,466.867 g.
2. To convert the mass of natural gas to volume, we can use the ideal gas law:
PV = nRT
Where:
P is the pressure in atm
V is the volume in liters
n is the number of moles
R is the ideal gas constant (0.0821 L.atm/mol.K)
T is the temperature in Kelvin
First, convert the temperature from °C to Kelvin:
T = 35 °C + 273.15 = 308.15 K
Next, we need to find the number of moles of methane using the mass we calculated in problem 1:
n = mass / molar mass
n = 39,466.867 g / 16 g/mol
n = 2466.679 mol
Now, we can use the ideal gas law to calculate the volume:
V = (n * R * T) / P
V = (2466.679 mol * 0.0821 L.atm/mol.K * 308.15 K) / 2 atm
V = 39,169.187 L
Therefore, the volume of natural gas required at 35°C and 2 atm is approximately 39,169.187 L.
I hope this helps! Let me know if you have any more questions.
Sure! I can help you with these calculations. Let's break it down step by step:
1. To calculate the mass of natural gas needed to heat the water, we first need to determine the change in temperature and the amount of heat required.
Step 1: Calculate the number of degrees Fahrenheit the water needs to be heated
Change in temperature = final temperature - initial temperature
Change in temperature = 98°F - 63°F = 35°F
Step 2: Convert Fahrenheit to Celsius
Since most of the calculations involved in this problem are based on the metric system, we need to convert the change in temperature to Celsius.
Temperature change (°C) = (Change in temperature - 32°F) × (5/9)
Temperature change (°C) = (35°F - 32°F) × (5/9)
Temperature change (°C) ≈ 1.67°C
Step 3: Calculate the amount of heat required to heat the water
The amount of heat required can be calculated using the formula:
Q = (mass of water) × (specific heat capacity of water) × (change in temperature)
The specific heat capacity of water is approximately 4.186 J/g°C.
We know the bathtub can hold 102 gallons of water, but we need to convert this to grams to be consistent with the units used for specific heat capacity (J/g°C). Here are the conversion factors:
1 gallon ≈ 3.78541 L
1 L ≈ 1000 g (since 1 L of water weighs approximately 1000 g)
Let's do the conversions:
102 gallons × 3.78541 L/gallon ≈ 385.635 L
385.635 L × 1000 g/L ≈ 385635 g
Now we can substitute the values into the formula:
Q = (mass of water) × (specific heat capacity of water) × (change in temperature)
Q ≈ 385635 g × 4.186 J/g°C × 1.67°C
After performing the calculation, you should obtain the amount of heat required to heat the water in joules.
Step 4: Convert heat to energy content of natural gas
To proceed, we need to know the energy content of methane when it is burned. The heat of combustion for methane is approximately 890 kJ/mol.
We can convert the amount of heat required from joules to kilojoules and then to moles of methane:
Heat (J) / 1000 ≈ Heat (kJ)
Heat (kJ) / (890 kJ/mol) ≈ moles of CH4
Step 5: Convert moles of CH4 to grams
The molar mass of methane (CH4) is approximately 16.04 g/mol.
Multiply the number of moles of CH4 by the molar mass to obtain the mass in grams.
And there you have it! By following these steps, you should be able to calculate the mass of natural gas needed to heat the water in a typical bathtub.
Let's move on to the second question:
2. To calculate the volume of natural gas at a different temperature and pressure, we can use the ideal gas law equation: PV = nRT.
Given:
Temperature (T) = 35°C = 35 + 273.15 K
Pressure (P) = 2 atm
To solve for the volume (V), we need to know the number of moles of CH4 (n) and the ideal gas constant (R). The molar mass of CH4 is approximately 16.04 g/mol.
Given the mass of natural gas obtained from the previous question, divide it by the molar mass to obtain the number of moles (n).
Now substitute the values into the ideal gas law equation:
(P)V = (n)(R)(T)
Since we are solving for V, rearrange the equation:
V = (n)(R)(T) / P
Substitute the values and calculate to obtain the volume of natural gas in liters.
I hope this explanation helps you understand how to approach these problems!
To solve these problems, we need to use the principles of stoichiometry and gas laws.
1. Calculating the mass of natural gas burned:
We know that the bathtub can hold 102 gallons of water, and we want to heat it from 63 ◦F to 98 ◦F. The difference in temperature is 98 - 63 = 35 ◦F.
To calculate the energy required to heat the water, we'll use the formula:
Energy = mass of water x specific heat capacity x temperature change
First, we need to find the mass of water in the bathtub. To do this, we'll convert the volume from gallons to liters and then to grams using the following conversion factors:
1 gallon = 3.78541 liters
1 liter of water = 1000 grams
So, the mass of water in the bathtub would be:
102 gallons × 3.78541 L/gallon × 1000 g/L = X grams
Next, we need to calculate the energy required to heat this mass of water. The specific heat capacity of water is 4.184 J/g°C.
Energy = X grams x 4.184 J/g°C x 35 ◦F
Now, we need to convert the energy from joules to the units of natural gas burned, which is grams of methane (CH4).
To do this, we need to use the balanced chemical equation for the combustion of methane:
CH4 + 2O2 → CO2 + 2H2O
From the equation, we can see that 1 mole of methane reacts to produce 1 mole of carbon dioxide (CO2). The molar mass of methane (CH4) is 16.04 g/mol.
So, we'll need to convert the energy (in joules) to moles of methane using the equation:
Energy (in joules) = moles of methane x enthalpy change
The enthalpy change for the combustion of methane is -890 kJ/mol.
After converting the energy to moles of methane, we can then convert the moles to grams using the molar mass of methane:
Moles of methane in grams = moles of methane x molar mass of methane
Finally, we'll have the mass of natural gas burned in grams.
2. Calculating the volume of natural gas:
To calculate the volume of natural gas at 35 ◦C and 2 atm, we'll use the ideal gas law equation:
PV = nRT
Where:
P = pressure (in atmospheres)
V = volume (in liters)
n = number of moles
R = ideal gas constant (0.0821 L·atm/K·mol)
T = temperature (in Kelvin)
To use this equation, we need to convert the temperature from 35 ◦C to Kelvin:
T (in K) = 35 ◦C + 273.15
Now, we can rearrange the ideal gas law equation to solve for V:
V = nRT / P
Substitute the given values for pressure, temperature, and R, and then solve for V in liters.
I hope this explanation helps!