By what factor does the rms speed of a molecule change if the temperature is increased from 10℃ to 1000℃?

Verified step by step guidance

1. Step 1: Understand the relationship between the root mean square (rms) speed of a molecule and temperature. The rms speed is given by the formula: $\sqrt{\frac{3}{k}}$, where $k$ is the Boltzmann constant, $T$ is the absolute temperature in Kelvin, and the factor of 3 comes from the degrees of freedom in the kinetic energy formula.
2. Step 2: Convert the given temperatures from Celsius to Kelvin. Use the formula: $T=T+273.15$. For 10℃, the temperature in Kelvin is $10+273.15$. For 1000℃, the temperature in Kelvin is $1000+273.15$.
3. Step 3: Recognize that the rms speed is proportional to the square root of the temperature. This means the factor by which the rms speed changes can be expressed as: $\frac{\sqrt{T}}{}$_finalT_initial, where $T$_final is the final temperature and $T$_initial is the initial temperature.
4. Step 4: Substitute the values of $T$_final and $T$_initial into the formula. For $T$_initial, use the Kelvin temperature corresponding to 10℃, and for $T$_final, use the Kelvin temperature corresponding to 1000℃.
5. Step 5: Simplify the expression to find the factor by which the rms speed changes. This involves calculating the square root of the ratio of the final temperature to the initial temperature.

Key Concepts

RMS Speed

The root mean square (RMS) speed of gas molecules is a measure of the average speed of particles in a gas. It is calculated using the formula v_rms = √(3kT/m), where k is the Boltzmann constant, T is the absolute temperature in Kelvin, and m is the mass of a molecule. This concept is crucial for understanding how temperature affects molecular motion.

Temperature and Kinetic Energy

Temperature is a measure of the average kinetic energy of the particles in a substance. As temperature increases, the kinetic energy of the molecules also increases, leading to higher speeds. This relationship is fundamental in thermodynamics and helps explain how changes in temperature influence molecular behavior.

Absolute Temperature Scale

The absolute temperature scale, measured in Kelvin (K), is essential for calculations involving gas laws and molecular speeds. To convert Celsius to Kelvin, you add 273.15. Understanding this scale is necessary for accurately determining the RMS speed when temperatures are given in Celsius.