Question

stars are surrounded by a white circle of gaseous material called the corona, which can be studied. the temperature of the corona is in the millions of degrees celsius, which is high enough to break up molecules and remove some or all of the electrons from atoms. one way astronomers have been able to estimate the temperature of the corona is by studying the emission lines of ions of certain elements. for example, the emission spectrum of an unknown element ion, here labeled as ( x^{15+} ), has been recorded and analyzed. given that it takes ( 4.3\times10^{4}\frac{kj}{mol} ) to convert ( x^{14+} ) to ( x^{15+} ), estimate the temperature of the stars corona. round your answer to 2 significant digits. (hint: the average kinetic energy of one mole of a gas is ( \frac{3}{2}rt ).) note: reference the fundamental constants table for additional information.

Explanation:

Step1: Convert energy unit

Given energy \(E = 4.3\times10^{4}\frac{kJ}{mol}=4.3\times 10^{7}\frac{J}{mol}\).

Step2: Use the formula for average kinetic energy

The formula for the average kinetic energy of one - mole of a gas is \(E=\frac{3}{2}RT\), where \(R = 8.314\frac{J}{mol\cdot K}\).
We need to solve for \(T\). Rearranging the formula \(T=\frac{2E}{3R}\).
Substitute \(E = 4.3\times 10^{7}\frac{J}{mol}\) and \(R = 8.314\frac{J}{mol\cdot K}\) into the formula:
\(T=\frac{2\times4.3\times 10^{7}}{3\times8.314}\)

$$
LATEXBLOCK0
$$

$$ T\approx3.4\times 10^{6}K $$

Answer:

\(3.4\times 10^{6}K\)