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Chapter 1: Problem 46

The temperature and pressure at the surface of Mars during a Martian spring day were determined to be \(-50^{\circ} \mathrm{C}\) and \(900 \mathrm{Pa}\). respectively. (a) Determine the density of the Martian atmosphere for these conditions if the gas constant for the Martian atmosphere is assumed to be equivalent to that of carbon dioxide. (b) Compare the answer from part (a) with the density of the Earth's atmosphere during a spring day when the temperature is \(18^{\circ} \mathrm{C}\) and the pressure \(101.6 \mathrm{kPa}(\mathrm{abs})\).

Short Answer

Expert verified
The density of the Martian atmosphere is about 0.0214 kg/m^3 and the density of the Earth's atmosphere is about 1.204 kg/m^3. So, the Earth's atmosphere is denser than the Martian atmosphere.

Step by step solution

01

Convert Temperatures to Kelvin

The temperature values are given in Celsius. We need to convert them to Kelvin before proceeding, using the formula \(K = °C + 273.15\). For Mars, the Kelvin temperature \(T_M\) is \(-50 + 273.15 = 223.15 K\). For Earth, the Kelvin temperature \(T_E\) is \(18 + 273.15 = 291.15 K\).
02

Apply the Ideal Gas Law to calculate the Martian atmosphere density

We are given that the value of the Gas Constant (\(R\)) for Mars is equivalent to that of carbon dioxide. Carbon dioxide has a gas constant of \(189 \mathrm{J / kg \cdot K}\). Substituting these values into the ideal gas law \(P = \rho R T\), we isolate the density \(\rho\) as follows, \(\rho_M = \frac{P_M}{R_M T_M}\). Substituting the given values, we get \(\rho_M = \frac{900}{189 \times 223.15} = 0.0214 \mathrm{kg/m^3}\)
03

Apply the Ideal Gas Law to calculate the Earth atmosphere density

The gas constant for air (\(R\)) on Earth is \(287 \mathrm{J / kg \cdot K}\). Also note that the pressure on Earth \(P_E = 101.6 kPa = 101600 Pa\). We use the modified equation \(\rho_E = \frac{P_E}{R_E T_E}\) to find the density. Applying the given values, we find \(\rho_E = \frac{101600}{287 \times 291.15} = 1.204 \mathrm{kg/m^3}\).
04

Compare the densities

We can directly compare the Martian and Earth densities to observe that the Earth's atmospheric density under the given conditions is greater than that of Mars'.

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