At some point in the combustion cycle of an engine, the temperature of a combusted air-fuel mixture is 582 degrees Celsius. Given that the pressure in the mixture is 6.2 bar, determine the density of the mixture (in kilogram per cubic metre).
Since we know the temperature and pressure but don't know the density, we can use the equation of state to solve this question. The only thing we need to do before applying this equation is convert the temperature to Kelvin and convert the pressure to Pascal (the SI units). This gives T=582+273.15=855.15K and p=6.2⋅105Pa. Now we can find the density:
p=ρRT
ρ=pRT=6.2⋅105287.00⋅855.15=2.526kg/m3
@anonymous p=rho×R×T
Therefore rho=p/RT
𝑝=𝜌𝑅𝑇
𝜌=𝑝𝑅𝑇=6.2⋅105287.00⋅855.15=2.526𝑘𝑔/𝑚3
T=582+273.15=855.15k
p=6.2
density(rho)=p/RT=6.2/(287*855.15)=2.52619
Why did the engine go to therapy?
Because it had too much pressure and needed to work on its density issues!
But seriously, to determine the density of the mixture, we can use the ideal gas law:
PV = nRT
where:
P = pressure (in bar) = 6.2 bar
V = volume (unknown for now, but let's assume 1 cubic meter for simplicity)
n = number of moles of gas (unknown for now, but we'll solve for it)
R = ideal gas constant (8.314 J/(mol·K), but let's use the value in SI units: 8.314 kg·m^2/(s^2·K·mol))
T = temperature (in Kelvin) = 582 + 273.15 K
Now let's solve for n:
n = PV / RT
Substituting the given values:
n = (6.2 bar * 1 m^3) / (8.314 kg·m^2/(s^2·K·mol) * (582 + 273.15) K)
Calculating this yields the number of moles of gas. Finally, we can calculate the density of the mixture by dividing the mass of the gas by its volume. The molar mass of the air-fuel mixture is needed for this calculation, which depends on the specific composition.
Now, what kind of gas goes to therapy? A therapist!
To determine the density of the mixture, we need to use the ideal gas law equation, which relates the pressure (P), temperature (T), and density (ρ) of a gas:
P = ρRT
Where:
P = Pressure in pascals (Pa)
ρ = Density in kilogram per cubic meter (kg/m³)
R = Gas constant (8.314 J/(mol·K))
T = Temperature in Kelvin (K)
First, we need to convert the temperature from degrees Celsius to Kelvin. The Kelvin temperature scale starts at absolute zero, where 0 °C is equal to 273.15 K.
T(K) = T(°C) + 273.15
T(K) = 582 °C + 273.15 = 855.15 K
Next, we need to convert the pressure from bar to pascals. 1 bar is equivalent to 100,000 Pascals (Pa).
P(Pa) = P(bar) * 100,000
P(Pa) = 6.2 bar * 100,000 = 620,000 Pa
Now we have the temperature (T) in Kelvin and the pressure (P) in Pascals. We can rearrange the ideal gas law equation to solve for the density (ρ):
ρ = P / (RT)
Substituting the given values:
ρ = 620,000 Pa / (8.314 J/(mol·K) * 855.15 K)
Simplifying the equation and canceling units:
ρ = 620,000 / (8.314 * 855.15) kg/m³
Calculating the result:
ρ ≈ 86.477 kg/m³
Therefore, the density of the mixture is approximately 86.477 kilogram per cubic meter (kg/m³).