Question

Calculate the root-mean-square speed of smoke particles of mass \(5.2 \times 10^{-14} \mathrm{~g}\) in air at \(14^{\circ} \mathrm{C}\) and \(1.07\) atm pressure.

Short answer

The root-mean-square speed of the smoke particles is approximately \(325.46 m/s\).

Step-by-step solution

Convert the Mass from Grams to Kilograms

The mass of smoke particles is given as \(5.2 \times 10^{-14}\) grams. To convert this to kilograms, multiply by \(1 \times 10^{-3} Kg/g\): \(5.2 \times 10^{-14} g \times 1 \times 10^{-3} Kg/g = 5.2 \times 10^{-17} Kg\)

Convert the Temperature from Celsius to Kelvin

The temperature is given as 14°C. To convert this to Kelvin, add 273.15: \(14°C + 273.15 = 287.15 K\)

Substitute Values Into the Speed Formula

Now, we can substitute the known values into the root-mean-square speed formula: \(V_{rms} = \sqrt{\frac{3(1.38 \times 10^{-23} J/K)(287.15 K)}{5.2 \times 10^{-17} Kg}}\)

Solve the Equation

Carry out the necessary calculations to solve for \(V_{rms}\). Be careful with the order of operations: multiplication and division first, then the square root. This yields a root-mean-square speed of approximately \(325.46 m/s\).

Key concepts

Mass Conversion

When calculating various scientific quantities, it’s crucial to have all values in consistent units. Often, mass is given in grams, but many formulas require the mass to be in kilograms. To convert mass from grams to kilograms, you need to multiply the value in grams by a conversion factor of

• \(1 \times 10^{-3}\) kilograms/gram.

This means that for every gram, there are

• \(0.001\) kilograms.

So, for a smoke particle with a mass of

• \(5.2 \times 10^{-14}\) grams,
• convert to kilograms by

\[5.2 \times 10^{-14} \text{ g} \times 1 \times 10^{-3} \text{ Kg/g} = 5.2 \times 10^{-17} \text{ Kg}\]This conversion ensures that the mass can be accurately used in subsequent calculations involving the root-mean-square speed formula.

Temperature Conversion

In physics problems, particularly those involving kinetic theory, it's critical to use the absolute temperature scale, Kelvin. Converting from Celsius to Kelvin is straightforward. The Kelvin scale is simply the Celsius scale shifted by 273.15 degrees.
To convert Celsius to Kelvin:

• Add 273.15 to the Celsius temperature.

So, for a given temperature of

• \(14^{\circ} \text{C}\),
• convert it to Kelvin using the formula

\[14^{\circ} \text{C} + 273.15 = 287.15 \text{ K}\]This conversion to Kelvin is necessary because the kinetic theory of gases relies on absolute temperature values to accurately calculate properties like speed.

Root-Mean-Square Speed Formula

The root-mean-square speed formula is a key concept in kinetic theory. It calculates the speed of particles in a gas. The formula is expressed as:\[V_{\text{rms}} = \sqrt{\frac{3kT}{m}}\], where:

• \(V_{\text{rms}}\) is the root-mean-square speed,
• \(k\) is the Boltzmann constant (\(1.38 \times 10^{-23} \text{ J/K}\),
• \(T\) is the temperature in Kelvin,
• \(m\) is the mass of a single particle in kilograms.

Using this formula, the speed can be determined by calculating the square root after substituting the known values of mass and temperature.
In the given problem, substituting

• \(T = 287.15 \text{ K}\)
• and \(m = 5.2 \times 10^{-17} \text{ Kg}\)
• into the formula yields

\[V_{\text{rms}} = \sqrt{\frac{3(1.38 \times 10^{-23} \text{ J/K})(287.15 \text{ K})}{5.2 \times 10^{-17} \text{ Kg}}}\]Proper use of this formula allows the computation of the average speed of gas particles.

Order of Operations in Calculations

Following the order of operations is crucial for solving mathematical equations correctly. The standard order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction), must be adhered to.
When solving for the root-mean-square speed, multiplication and division are performed first within the fraction:

• Calculate the product of the constants, the Boltzmann constant \(k\), and the temperature \(T\).
• Divide this product by the mass \(m\).
• Finally, take the square root of the resulting value to find \(V_{\text{rms}}\).

This meticulous approach ensures that each operation is executed correctly, leading to the accurate calculation of the root-mean-square speed, which we find to be approximately

• \(325.46 \text{ m/s}\).

Being precise with the order of operations is critical in achieving the correct result in scientific calculations.