Question

The number density of gas atoms at a certain location in the space above our planet is about 1.05 × 1011 m-3, and the pressure is 2.7 × 10-10 Pa in this region.What is the temperature there?

Answer 1

To solve this problem we will apply the concepts given by the ideal gas equation, which mathematically can be described as

$PV = NkT$

Here

P = Pressure

V = Volume

N = Number of atoms of molecules

k = Bolzmann constant

T = Temperature

Rearranging to find the temperature we have

$T = \frac{PV}{Nk}$

Since the value given in the exercise is a unit of atoms per volume, we will readjust the equation like this

$T = \frac{P}{\frac{N}{V}k}$

Replacing we have,

$T = \frac{(2.7*10^{-10}N/m^2)}{(1.05*10^{11}/m^3)(1.38*10^{-23}J/K)}$

$T = 186.3K$

$T = -86.81\°C$

Therefore the temperature is -86.81°C or 186.3K

Answer 2

The temperature of the gas at the given region is 186.34 K.

The given parameters;

• specific density, n/V = 1.05 x 10¹¹ m⁻³
• pressure, P = 2.7 x 10⁻¹⁰ Pa

The temperature of the gas at the given region is calculated as follows;

$PV = nkT\\\\T = \frac{PV}{nk} \\\\T = \frac{P}{\frac{n}{V} k}$

where;

k is Boltzmann constant = 1.38 x 10⁻²³ J/K

$T = \frac{2.7 \times 10^{-10}}{1.05 \times 10^{11} \times 1.38 \times 10^{-23}} \\\\T = 186.34 \ K$

Thus, the temperature of the gas at the given region is 186.34 K.

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