a very small gas bubble seen in a microscope has a diameter of 1 micrometer. If the pressure is one bar and the temperature is 25 degrees celsius calculate the number of gas molecules in the bubble
Convert a diameter of 1 micrometer to volume in liters. volume = (4/3)*pi*r^3. Hint:If you use radius in cm, the volume will be in cc (cubic centimeters) and 1000 cc = 1L.
Convert 1 bar to atmospheres and use PV = nRT and solve for n.
A mole contains 6.022E23 molecules. Solve for # molecules.
thank you
To calculate the number of gas molecules in a bubble, we can use the ideal gas law equation, which relates the pressure, volume, temperature, and the number of gas molecules (moles).
The ideal gas law equation is given by:
PV = nRT
Where:
P = Pressure (in Pa)
V = Volume (in m³)
n = Number of moles of gas
R = Ideal gas constant (8.314 J/(mol·K))
T = Temperature (in Kelvin)
First, we need to convert the temperature from Celsius to Kelvin:
T(K) = T(°C) + 273.15
T(K) = 25 °C + 273.15 = 298.15 K
Given:
Diameter of the bubble (d) = 1 micrometer
The volume (V) of a spherical bubble can be calculated using the equation:
V = (4/3) * π * (r^3)
Where:
r = Radius of the bubble (d/2)
Let's calculate the volume of the bubble:
d = 1 micrometer = 1 × 10^(-6) m
r = d/2 = (1 × 10^(-6))/2 = 0.5 × 10^(-6) m
V = (4/3) * π * (0.5 × 10^(-6))^3
Now, we can calculate the number of moles (n):
n = (PV) / (RT)
Since the pressure and volume given are in bar and micrometers respectively, we need to convert them to Pascal and meters:
P(bar) = 1 bar = 100,000 Pa
V(μm³) = V(m³) * 10^(-18)
P(Pa) = 100,000 Pa
V(m³) = V(μm³) * 10^(-18)
R = 8.314 J/(mol·K)
T(K) = 298.15 K
Now, let's substitute these values into the equation to calculate the number of moles (n):
n = (P * V) / (R * T)
Afterward, we can use Avogadro's number (6.022 × 10^23) to convert moles into the number of gas molecules.
Finally, the number of gas molecules in the bubble is given by:
Number of gas molecules = n * Avogadro's number
To calculate the number of gas molecules in the bubble, we can use the ideal gas law equation:
PV = nRT
Where:
- P is the pressure of the gas (in bar)
- V is the volume of the gas (in cubic meters)
- n is the number of gas molecules
- R is the ideal gas constant (0.0831 bar⋅m^3/(mol⋅K))
- T is the temperature in Kelvin (25 degrees Celsius = 25 + 273 = 298 K)
First, we need to calculate the volume of the gas bubble:
V = (4/3) * π * r^3
Where:
- V is the volume of the gas bubble
- π is a constant (approximately 3.14159)
- r is the radius of the gas bubble (diameter/2)
In this case, the diameter is given as 1 micrometer, so the radius would be 0.5 micrometers or 0.5 × 10^(-6) meters.
Let's plug in the values and calculate step by step:
1. Convert the diameter to meters:
radius = (1 × 10^(-6)) / 2 = 0.5 × 10^(-6) meters
2. Calculate the volume of the gas bubble:
V = (4/3) * π * (0.5 × 10^(-6))^3 = (4/3) * π * (0.125 × 10^(-18)) m^3
3. Convert the volume from cubic meters to liters (1 m^3 = 1000 liters):
V = (4/3) * π * (0.125 × 10^(-18)) * 1000 = (4/3) * π * (1.25 × 10^(-15)) liters
Now, we have the volume of the gas bubble (V) in liters.
4. Substitute the values into the ideal gas law equation:
PV = nRT
(1 bar) * (V liters) = (n) * (0.0831 bar⋅m^3/(mol⋅K)) * (298 K)
V = n * 25.9488
Now, rearrange the equation to solve for n (number of gas molecules):
n = (V / 25.9488)
Substitute the value of V into the equation:
n = ((4/3) * π * (1.25 × 10^(-15)) / 25.9488)
Finally, calculate the value of n using a calculator:
n = 1.54 × 10^(-21) molecules
Therefore, the number of gas molecules in the bubble is approximately 1.54 × 10^(-21) molecules.