Question

The root-mean-square speed of a certain gaseous oxide is \(493 \mathrm{~m} / \mathrm{s}\) at \(20^{\circ} \mathrm{C}\). What is the molecular formula of the compound?

Short answer

The molecular formula of the compound is NO (Nitric Oxide).

Step-by-step solution

Understand the Root-Mean-Square Speed Formula

The root-mean-square speed of gas molecules is given by the formula:\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]where \( v_{rms} \) is the root-mean-square speed, \( R \) is the universal gas constant \( 8.314 \text{ J/mol K} \), \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas in kg/mol.

Convert the Temperature to Kelvin

First, convert the temperature from Celsius to Kelvin using the formula:\[ T(K) = T(^{\circ}C) + 273.15 \]For \(20^{\circ}C\), this becomes:\[ T = 20 + 273.15 = 293.15 \text{ K} \]

Rearrange the Formula to Solve for Molar Mass

Rearrange the root-mean-square speed formula to solve for \( M \) (molar mass):\[ M = \frac{3RT}{v_{rms}^2} \]Substitute \( R = 8.314 \text{ J/mol K} \), \( T = 293.15 \text{ K} \), and \( v_{rms} = 493 \text{ m/s} \) into the equation.

Calculate the Molar Mass

Substitute the values into the rearranged equation:\[ M = \frac{3 \times 8.314 \times 293.15}{493^2} \]Calculate the value:\[ M = \frac{7304.3647}{243049} \approx 0.03005 \text{ kg/mol} \]Convert \( M \) to grams per mole:\[ M = 30.05 \text{ g/mol} \]

Determine the Molecular Formula

The molar mass calculated is approximately 30 g/mol. Considering it is a gaseous oxide, possible diatomic oxides with a similar molar mass include NO (Nitric Oxide) and CH2O (formaldehyde, though not typically gaseous at room temperature). Therefore, the molecular formula is likely \( \text{NO} \), which has a molar mass of approximately 30 g/mol.

Key concepts

Molar Mass Calculation

The calculation of molar mass is key to determining the composition of a compound. The molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It provides insight into the weight of a compound relative to the number of particles it contains. To calculate it using root-mean-square speed, we rearrange the rms speed formula: \[ M = \frac{3RT}{v_{rms}^2} \] With this formula, we input relevant values for the universal gas constant, temperature in Kelvin, and the rms speed of the gas. This efficiency of calculation allows chemists to predict molecular composition without directly weighing the complex atoms involved. Once we determine the molar mass, we compare it to known values to estimate the compound's identity.

Universal Gas Constant

The universal gas constant, often symbolized by \( R \), is a critical constant in the study of gases and their behaviors. It features prominently in the calculation of the root-mean-square speed among other gas laws. - The value is \( 8.314 \text{ J/mol K} \). - This constant links the amount of gas with its energy and temperature, aiding in calculating energy per mole using temperature and pressure. The consistent use of \( R \) in formulas such as the Ideal Gas Law (\[ PV = nRT \]) illustrates its importance in converting physical observations of gases to standardized equations, thereby making universal comparisons across different conditions.

Temperature Conversion

Understanding temperature conversion is essential in scientific calculations involving gases. The conversion from Celsius to Kelvin is particularly important because many gas law equations demand temperature in Kelvin to ensure accuracy and consistency.- The formula to convert is: \[ T(K) = T(^{\circ}C) + 273.15 \] - For example, converting \( 20^{\circ}C \) to Kelvin results in \( 293.15 \text{ K} \).This conversion adjusts for absolute zero, where 0 Kelvin is the theoretical point of no particle energy, unlike the arbitrary zero starting point of the Celsius scale. By using Kelvin, scientists maintain precision in thermal calculations that might otherwise be skewed by relative differences in temperature scales.

Gas Molecular Formula

The molecular formula of a gas provides critical information about the types and numbers of atoms in a molecule. It offers insights into the molecular structure and its potential chemical behavior. In the context of the root-mean-square speed problem, determining the molecular formula involves calculating the molar mass and linking this to known substances. - For gaseous oxides with a molar mass near 30 g/mol, plausible candidates include nitric oxide (NO) or other similar low molecular weight compounds. - The criteria for being considered gaseous at room temperature further refines possible candidates. By connecting the estimated molar mass to these known compounds, chemists can hypothesize the molecular formula of the unknown substance, aiding in further analysis or experimental planning.