calculate the minimum number of grams of C3H8(g) that must be combusted to provide the energy necessary to convert 3.36 kg of H2O from its solid form at -10.0°C to its liquid form at 80.0°C
To solve this problem, we are going to find out how much energy is required to convert the ice at -10.0°C to liquid water at 80.0°C in two steps:
1. Heating the ice from -10.0°C to 0.0°C.
2. Melting the ice and subsequently heating the liquid to 80.0°C
For step 1:
Using the heat capacity of ice (Cice) = 2.093 J/g°C, we can calculate the energy required as follows:
q1 = mass * Cice * ΔT
where mass = 3.36 kg = 3360 g, ΔT = 10.0°C
q1 = 3360g * 2.093 J/g°C * 10.0°C
q1 = 70312.8J
For step 2:
Melting the ice and heating the liquid water to 80.0°C.
Melting the ice at 0°C to water will require the energy (q2) as follows:
q2 = mass * Lf
where Lf (latent heat of fusion) = 334 J/g
q2 = 3360g * 334 J/g
q2 = 1122240J
Heating the water from 0°C to 80°C will require energy (q3) as follows:
Using the heat capacity of water (Cwater) = 4.186 J/g°C
q3 = mass * Cwater * ΔT
ΔT = 80°C
q3 = 3360g * 4.186 J/g°C * 80.0°C
q3 = 1123555.2J
Total energy required (q) = q1 + q2 + q3 = 70312.8J + 1122240J + 1123555.2J = 2316107.8 J.
Now we need to find the amount of C3H8 combustion needed to produce that energy. The combustion reaction for propane is:
C3H8(g) + 5O2(g) → 3CO2(g) + 4H2O(l)
1 mole propane combustion gives ΔH = -2220 kJ/mol
Energy produced in the reaction = -ΔH = 2220 kJ/mol
We need to find out how many moles of C3H8 are required to produce 2316107.8 J.
Moles of propane = Energy required / Energy produced per mole
= 2316107.8 J / 2220 kJ/mol
= 2316107.8 J / (2220 * 1000) J/mol
= 1.04260 mol of propane
The molar mass of propane (C3H8) is 3(12.01) + 8(1.008) = 44.096 g/mol.
So the mass of C3H8 required is mass = moles * Molar mass
= 1.04260 mol * 44.096 g/mol
= 45.98 g (approximately)
The minimum number of grams of C3H8(g) that must be combusted to provide the energy necessary to convert 3.36 kg of H2O from its solid form at -10.0°C to its liquid form at 80.0°C is approximately 45.98 g.
To calculate the minimum number of grams of C3H8(g) required to provide the energy necessary for this phase change, we need to consider the heat required to convert the ice to liquid water.
The total heat required can be calculated using the following equation:
Q = m * ΔH
Where:
Q = heat (in Joules)
m = mass of H2O (in kg)
ΔH = heat of fusion (in J/kg)
The heat of fusion for water is 334,000 J/kg.
First, let's calculate the heat required to melt the ice:
Q1 = m1 * ΔH1
Where:
m1 = mass of ice (in kg)
ΔH1 = heat of fusion (334,000 J/kg)
The mass of ice can be calculated using the density of ice:
density = mass / volume
The density of ice is approximately 917 kg/m³.
The volume of ice can be calculated using the formula:
volume = mass1 / density1
Now we can calculate the heat required to melt the ice:
Q1 = m1 * ΔH1
= (volume1 * density1) * ΔH1
Next, let's calculate the heat required to heat the liquid water from -10.0°C to 80.0°C:
Q2 = m2 * c * ΔT
Where:
m2 = mass of liquid water (in kg)
c = specific heat capacity of water (4,186 J/kg°C)
ΔT = change in temperature (80.0°C - (-10.0°C))
Now we can calculate the mass of liquid water:
m2 = volume2 * density2
To find the total heat required, we sum up Q1 and Q2:
Q_total = Q1 + Q2
Finally, we calculate the minimum amount of C3H8(g) required by using the heat of combustion for C3H8(g), which is -2220 kJ/mol:
Moles = Q_total / heat of combustion
Moles = (Q_total / heat of combustion) / 1000 [to convert kJ to J]
Mass of C3H8(g) = Moles * molar mass of C3H8(g)
Now let's plug in the values and calculate step by step using the given data:
Volume1 = m1 / density1 = 3.36 kg / 917 kg/m³ = 0.003664 m³
Q1 = (volume1 * density1) * ΔH1 = (0.003664 m³ * 917 kg/m³) * 334,000 J/kg = 1,063.46 kJ
volume2 = m2 / density2 = 3.36 kg / 1000 kg/m³ = 0.00336 m³
Q2 = m2 * c * ΔT = (0.00336 m³ * 1000 kg/m³) * 4,186 J/kg°C * 90°C = 1,099,692.8 J
Q_total = Q1 + Q2 = 1,063.46 kJ + 1,099,692.8 J = 1,100,756.26 J
Moles = (Q_total / heat of combustion) / 1000 = (1,100,756.26 J / -2220 kJ/mol) / 1000 = -0.496 mol
Mass of C3H8(g) = Moles * molar mass of C3H8(g) = -0.496 mol * 44.1 g/mol = -21.8616 g
Since mass cannot be negative, it means that no C3H8(g) is required for this phase change.
To calculate the minimum number of grams of C3H8(g) required to provide the necessary energy, we need to follow these steps:
1. Determine the amount of heat required to convert 3.36 kg of H2O from solid to liquid form.
To find the amount of heat required, we can use the formula:
Q = m × c × ΔT
Where:
Q = Heat energy (in Joules)
m = Mass of the substance (in kg)
c = Specific heat capacity (in J/kg°C)
ΔT = Change in temperature (in °C)
For converting solid H2O to liquid H2O, we need to consider the heat required for two processes: raising the temperature from -10.0°C to 0°C, and then from 0°C to 80.0°C. The specific heat capacity values we'll use are:
For ice (solid H2O): c = 2.09 J/g°C
For water (liquid H2O): c = 4.18 J/g°C
2. Convert the mass of H2O from kg to grams.
3. Calculate the total heat required.
For the first stage, from -10.0°C to 0°C:
Q1 = m × c(ice) × ΔT1
Where ΔT1 = (0 - (-10.0))
For the second stage, from 0°C to 80.0°C:
Q2 = m × c(water) × ΔT2
Where ΔT2 = 80.0 - 0
4. Add both heat values (Q1 + Q2) to get the total heat required.
5. Calculate the heat released during the combustion of C3H8(g).
The balanced chemical equation for the combustion of C3H8(g) is:
C3H8(g) + 5O2(g) → 3CO2(g) + 4H2O(g)
The standard enthalpy of combustion (∆H) for C3H8(g) is -2220 kJ/mol.
We can calculate the heat released by considering the molar mass of C3H8 and the enthalpy of combustion per mole of C3H8.
6. Convert the heat released from kJ to J.
7. Calculate the minimum number of moles of C3H8(g) required.
By using the heat values obtained in step 4 and step 6 and the enthalpy of combustion, we can calculate the number of moles of C3H8 required.
8. Convert moles of C3H8(g) to grams.
9. Calculate the minimum number of grams of C3H8(g) required to provide the necessary energy.
By multiplying the moles of C3H8 calculated in step 7 by the molar mass of C3H8, we can obtain the minimum number of grams required.
Using these steps, you can calculate the minimum number of grams of C3H8(g) needed to provide the necessary energy to convert H2O from solid to liquid form.