Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 18: Thermal Properties of Matter

Problem 18.30b

Chapter 18, Problem 18.30b

Martian Climate. The atmosphere of Mars is mostly CO2 (molar mass 44.0 g/mol) under a pressure of 650 Pa, which we shall assume remains constant. In many places the temperature varies from 0.0°C in summer to -100°C in winter. Over the course of a Martian year, what are the ranges of (b) the density (in mol/m^3) of the atmosphere?

Verified step by step guidance

First, understand that the ideal gas law is applicable here, which is given by the equation:

P V = n R T

, where

P

is the pressure,

V

is the volume,

n

is the number of moles,

R

is the ideal gas constant, and

T

is the temperature in Kelvin.

Convert the given temperatures from Celsius to Kelvin. The conversion formula is

T

_k = T_c + 273.15. So, for summer,

0.0 ° C

becomes

273.15 K

, and for winter,

-100 ° C

becomes

173.15 K

Rearrange the ideal gas law to solve for the number of moles per unit volume (density in mol/m³):

n / V = P /(R T)

. Here,

P = 650 Pa

and

R = 8.314 J /(mol K)

Calculate the density for the summer temperature: Substitute

T = 273.15 K

into the rearranged equation to find

n / V

for summer.

Calculate the density for the winter temperature: Substitute

T = 173.15 K

into the rearranged equation to find

n / V

for winter. This will give you the range of densities over the Martian year.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physics that relates the pressure, volume, temperature, and amount of gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law is crucial for calculating the density of gases under varying conditions.

Temperature Conversion

Temperature conversion is essential when working with the Ideal Gas Law, as it requires temperature in Kelvin. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. For example, 0°C is 273.15 K, and -100°C is 173.15 K. Accurate conversion ensures correct calculations in thermodynamic equations.

Density of a Gas

The density of a gas is defined as its mass per unit volume, often expressed in mol/m^3 for gases. Using the Ideal Gas Law, density can be calculated as n/V = P/RT, where n/V is the molar density. Understanding how pressure and temperature affect density is key to solving problems involving atmospheric conditions on planets like Mars.

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