The solar corona is a very hot atmosphere surrounding the visible surface of the sun. X-ray emissions from the corona show that its temperature is about 2×106 K. The gas pressure in the corona is about 0.03 Pa. Estimate the number density of particles in the solar corona.

Start by identifying the ideal gas law, which relates pressure, number density, temperature, and the Boltzmann constant: P = nkT, where P is the pressure, n is the number density, k is the Boltzmann constant, and T is the temperature. Rearrange the ideal gas law to solve for the number density n: n = PkT. Substitute the given values into the equation: P = 0.03 Pa, T = 2 × $$10^{6} K$$, and k = 1.38 × $$10^{-23} J/K ($$Boltzmann constant). Perform the substitution: n = 0.031.38 × $$10^{-23} × 2$$ × $$10^{6}. $$Simplify the expression to calculate the number density n. This will give the estimated number of particles per cubic meter in the solar corona.

Ch 18: A Macroscopic Description of Matter

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 18: A Macroscopic Description of Matter

Problem 24

Chapter 18, Problem 24

The solar corona is a very hot atmosphere surrounding the visible surface of the sun. X-ray emissions from the corona show that its temperature is about 2×10⁶ K. The gas pressure in the corona is about 0.03 Pa. Estimate the number density of particles in the solar corona.

Verified step by step guidance

Start by identifying the ideal gas law, which relates pressure, number density, temperature, and the Boltzmann constant:

P = n k T

, where

P

is the pressure,

n

is the number density,

k

is the Boltzmann constant, and

T

is the temperature.

Rearrange the ideal gas law to solve for the number density

n

n = \frac{P}{k}

Substitute the given values into the equation:

P = 0.03

Pa,

T = 2 × 10

⁶ K, and

k = 1.38 × 10

^-23 J/K (Boltzmann constant).

Perform the substitution:

n = \frac{0.03}{1.38 × 10}

^-23 × 2 × 10⁶.

Simplify the expression to calculate the number density

n

. This will give the estimated number of particles per cubic meter in the solar corona.

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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 relates pressure, volume, temperature, and number of particles in a 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 allows us to estimate the number density of particles by rearranging the equation to find n/V, which is crucial for understanding the conditions in the solar corona.

Number Density

Number density is defined as the number of particles per unit volume, typically expressed in particles per cubic meter. It provides insight into the concentration of particles in a given space, which is essential for analyzing the properties of gases, including those in the solar corona. By using the Ideal Gas Law, we can calculate the number density from the known pressure and temperature of the corona.

Thermal Equilibrium

Thermal equilibrium occurs when a system's temperature is uniform throughout, meaning that energy is distributed evenly among particles. In the context of the solar corona, the high temperature of about 2×10^6 K indicates that the particles are in rapid motion, contributing to the gas pressure. Understanding thermal equilibrium helps explain the behavior of the corona's plasma and its interactions with electromagnetic radiation, such as X-rays.

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