# The density of a gas is 1.96 g L–1 at 1.00 atm and 0

°C. What is the density of this gas at 0.855 atm and 25.0
C?

## Two ways to do this. The long way, but easier to explain, and the short way but harder to explain. The long way first:

The general gas formula can be modifed to
P*M = dRT where M is molar mass and d is density in g/L. Substitute into that and solve for M. Then use the same equation, substitute the new conditions and solve for the new density.

Shorter way.
1.96 x (pres factor) x (Temp factor) =
1.96 g/L x (0.855/1) x (273/298) = ?
How does the shorter way work?
P factor. P decreases therefore L increases and density gets smaller so the factor must be less than 1 which makes the small number go in the numerator.
T factor. T increases from 273 to 298, L must increase, larger L means smaller density so the T factor must be less than 1 which makes the small number go on top.

## To determine the density of a gas at different conditions, you can use the ideal gas law equation and apply the concept of molar volume. The ideal gas law equation is as follows:

PV = nRT

Where:
P = pressure (in atm)
V = volume (in L)
n = number of moles
R = ideal gas constant (0.0821 L·atm/mol·K)
T = temperature (in Kelvin)

To find the density of the gas, we can rearrange the ideal gas law equation as follows:

density (d) = (molar mass of the gas) / (molar volume)

Step 1: Calculate the molar volume at the given conditions.
To convert the given temperature of 0°C to Kelvin:
T1 = 0°C + 273.15 = 273.15 K (since 0°C is equivalent to 273.15 K)

The given conditions are:
P1 = 1.00 atm
T1 = 273.15 K
d1 = 1.96 g/L

Using the ideal gas law equation, we can solve for the molar volume (V1):

V1 = (n * R * T1) / P1

Step 2: Calculate the number of moles (n) at the given conditions:
We know the mass, molar mass, and volume. From that, we can calculate the number of moles (n):

n = (mass) / (molar mass)
= (density * volume) / (molar mass)
= (d * V1) / (molar mass)

Step 3: Calculate the molar volume (V2) at the new conditions:
For the new conditions, we have:
P2 = 0.855 atm
T2 = 25.0°C + 273.15 = 298.15 K (since 25.0°C is equivalent to 298.15 K)

Using the ideal gas law equation, we can solve for the molar volume (V2):

V2 = (n * R * T2) / P2

Step 4: Calculate the density at the new conditions:
Using the rearranged equation for density:
density (d2) = (molar mass) / (molar volume)
d2 = (molar mass) / V2

Now you can substitute V2 using the calculated value and solve for d2.

It is essential to know the molar mass of the gas to obtain the final answer.