Ch 18: A Macroscopic Description of Matter

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 18: A Macroscopic Description of Matter

Problem 47

Chapter 18, Problem 47

On average, each person in the industrialized world is responsible for the emission of 10,000 kg of carbon dioxide (CO₂) every year. This includes CO₂ that you generate directly, by burning fossil fuels to operate your car or your furnace, as well as CO₂ generated on your behalf by electric generating stations and manufacturing plants. CO₂ is a greenhouse gas that contributes to global warming. If you were to store your yearly CO₂ emissions in a cube at STP, how long would each edge of the cube be?

Verified step by step guidance

Step 1: Start by recalling the ideal gas law, which is given by the equation:

P V = n R T

. Here,

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. At standard temperature and pressure (STP),

P

is 1 atm,

T

is 273.15 K, and

R

is 0.0821 L·atm/(mol·K).

Step 2: Calculate the number of moles of CO₂ emitted in a year. Use the molar mass of CO₂, which is approximately 44 g/mol. Convert the yearly emission of 10,000 kg of CO₂ to grams (1 kg = 1000 g), and then divide by the molar mass to find the number of moles:

n = \frac{10,000 \times 1000}{44}

Step 3: Rearrange the ideal gas law to solve for the volume

V

V = \frac{n R T}{P}

. Substitute the values for

n

(calculated in Step 2),

R

= 0.0821 L·atm/(mol·K),

T

= 273.15 K, and

P

= 1 atm.

Step 4: Once the volume

V

is determined, recall that the volume of a cube is given by

V = s^3

, where

s

is the length of one edge of the cube. Solve for

s

by taking the cube root of the volume:

s = V^{\frac{1}{3}}

Step 5: Substitute the calculated volume from Step 3 into the cube root formula to find the edge length of the cube. Ensure the units are consistent (e.g., convert the volume to cubic meters if necessary) to express the edge length in meters.

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Key Concepts

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

Greenhouse Gases

Greenhouse gases, such as carbon dioxide (CO₂), trap heat in the Earth's atmosphere, leading to the greenhouse effect. This process is essential for maintaining the planet's temperature but excessive emissions from human activities enhance this effect, contributing to global warming and climate change.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a reference point used in chemistry and physics, defined as 0 degrees Celsius (273.15 K) and 1 atmosphere of pressure. At STP, one mole of an ideal gas occupies 22.4 liters, which is crucial for calculating the volume of gases and understanding how to convert mass of CO₂ emissions into a physical volume.

Volume of a Cube

The volume of a cube is calculated using the formula V = s³, where 's' is the length of one edge. To find the length of each edge when given the total volume, one can rearrange the formula to s = V^(1/3). This concept is essential for determining the dimensions of the cube that would contain the total CO₂ emissions.

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