At the top of mount everest the density of air is 1.2kilogram/meter cube and its pressure and temperature are 200kpa & 150k.what is the volume of the air at the bottom of mount everest if p=300pa & T=400k ?
To find the volume of the air at the bottom of Mount Everest, we can use the ideal gas law, which states that the product of pressure, volume, and temperature is proportional to the number of gas molecules present. The formula for the ideal gas law is:
PV = nRT
Where:
- P is the pressure of the gas in Pascals (Pa)
- V is the volume of the gas in cubic meters (m³)
- n is the number of gas molecules (constant for a given amount of gas)
- R is the ideal gas constant (8.3145 J/(mol·K))
- T is the temperature of the gas in Kelvin (K)
First, we need to convert the given pressure and temperature values at the top and bottom of Mount Everest to their respective SI units:
At the top of Mount Everest:
- Pressure (P) = 200 kPa = 200,000 Pa
- Temperature (T) = 150 K
At the bottom of Mount Everest:
- Pressure (P) = 300 Pa
- Temperature (T) = 400 K
Now, let's plug in the values into the ideal gas law equation and solve for the volume at the bottom of Mount Everest:
(V₁ × P₁) / T₁ = (V₂ × P₂) / T₂
Where:
- V₁ is the volume at the top of Mount Everest
- P₁ is the pressure at the top of Mount Everest
- T₁ is the temperature at the top of Mount Everest
- V₂ is the volume at the bottom of Mount Everest (what we want to find)
- P₂ is the pressure at the bottom of Mount Everest
- T₂ is the temperature at the bottom of Mount Everest
Substituting the given values:
(V₁ × 200,000) / 150 = (V₂ × 300) / 400
To find V₂, we can rearrange the equation to isolate it on one side:
V₂ = (V₁ × 200,000 × 400) / (150 × 300)
Now, we need to find the ratio between the densities of air at the top and bottom of Mount Everest. Density (ρ) is given by the equation:
ρ = mass / volume
At the top of Mount Everest, the density is 1.2 kg/m³. We need to find the density at the bottom.
Given that the density at the top (ρ₁) is 1.2 kg/m³ and the pressure and temperature at the bottom (P₂ and T₂) are given, we can use the ideal gas law once again to find the number of molecules (n₂) at the bottom:
n₂ = (P₂ × V₂) / (R × T₂)
Using the ratio between the densities, we can then find the volume (V₂) at the bottom:
V₂ = (ρ₁ × V₁) / ρ₂
Now, we can substitute the values we have and calculate the volume (V₂) at the bottom of Mount Everest.