Liquid sodium is being considered as an engine coolant. How many grams of liquid sodium (minimum) are needed to absorb 3.60 MJ of energy (in the form of heat) if the temperature of the sodium is not to increased by more than 10.0 °C? Use CP = 30.8 J/(K·mol) for Na(l) at 500 K.
3.6E6 = mols Na x specific heat Na x 10
Solve for mols Na and convert to grams. g = mols x molar mass.
I assume 3.60 MJ stands for 3.60 megajoules. If not you will need t adjust the factor I used above.
To calculate the amount of liquid sodium needed to absorb 3.60 MJ of energy without increasing the temperature by more than 10.0 °C, we'll follow these steps:
Step 1: Convert 3.60 MJ to joules:
1 MJ = 1,000,000 J
3.60 MJ = 3,600,000 J
Step 2: Determine the molar heat capacity (CP) of liquid sodium:
CP = 30.8 J/(K·mol)
Step 3: Calculate the change in temperature (ΔT):
ΔT = 10.0 °C = 10 K
Step 4: Use the formula to calculate the number of moles (n) of liquid sodium needed:
n = Q / (CP * ΔT)
where Q is the amount of energy in joules, and ΔT is the change in temperature in Kelvin.
n = 3,600,000 J / (30.8 J/(K·mol) * 10 K)
Step 5: Simplify the equation:
n = 3,600,000 J / 308 J/mol
Step 6: Calculate the number of moles:
n ≈ 11,688.31 mol
Step 7: Convert moles to grams:
The molar mass of sodium (Na) is approximately 22.99 g/mol.
grams = n * molar mass
grams ≈ 11,688.31 mol * 22.99 g/mol
grams ≈ 268,689.72 g
Therefore, a minimum of approximately 268,690 grams (or about 268.69 kg) of liquid sodium is needed to absorb 3.60 MJ of energy without increasing the temperature by more than 10.0 °C.
To calculate the mass of liquid sodium needed to absorb a certain amount of energy while maintaining a specific temperature change, we can use the equation:
q = m * Cp * ΔT
where:
q = amount of heat absorbed (in J)
m = mass of liquid sodium (in grams)
Cp = molar heat capacity of sodium (in J/(K·mol))
ΔT = temperature change (in K)
First, convert the given energy in MJ to J:
3.60 MJ = 3.60 × 10^6 J
Next, rearrange the equation to solve for m:
m = q / (Cp * ΔT)
Now, substitute the known values:
q = 3.60 × 10^6 J
Cp = 30.8 J/(K·mol)
ΔT = 10.0 °C = 10.0 K
m = (3.60 × 10^6 J) / (30.8 J/(K·mol) * 10.0 K)
m = (3.60 × 10^6 J) / (308 J/mol)
Now, we need to convert from moles to grams using the molar mass of sodium:
molar mass of sodium (Na) = 22.99 g/mol
m = (3.60 × 10^6 J) / (308 J/mol) * (22.99 g/mol)
m ≈ 258,370 g
Therefore, a minimum of approximately 258,370 grams of liquid sodium is needed to absorb 3.60 MJ of energy with a temperature increase of no more than 10.0 °C.