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How many molecules are there in \(70 \mathrm{~mL}\) of benzene? For benzene, \(\rho=0.88 \mathrm{~g} / \mathrm{cm}^{3}\) and \(M=78 \mathrm{~kg} / \mathrm{kmol}\). Remember that \(1 \mathrm{~g} / \mathrm{cm}^{3}=1000 \mathrm{~kg} / \mathrm{m}^{3}\) and so here \(\rho=880 \mathrm{~kg} / \mathrm{m}^{3}\). $$ \begin{aligned} \text { Mass of } 70 \mathrm{~cm}^{3} &=m=\rho V=\left(880 \mathrm{~kg} / \mathrm{m}^{3}\right)\left(70 \times 10^{-6} \mathrm{~m}^{3}\right)=0.0616 \mathrm{~kg} \\ m_{0} &=\frac{M}{N_{A}}=\frac{78 \mathrm{~kg} / \mathrm{kmol}}{6.02 \times 10^{26 \mathrm{kmol}^{-1}}}=1.30 \times 10^{-25} \mathrm{~kg} \\ \text { Number in } 70 \mathrm{~cm}^{3} &=\frac{m}{m_{0}}=\frac{0.0616 \mathrm{~kg}}{1.30 \times 10^{-25} \mathrm{~kg}}=4.8 \times 10^{23} \end{aligned} $$

Step by step solution

01

Convert Volume

We start by converting the given volume from milliliters (mL) to cubic meters (m鲁), since the density is given in kg/m鲁. Given that 1 mL = 1 cm鲁 and 1 cm鲁 = 1脳10鈦烩伓 m鲁, the volume is:\[ V = 70 ext{ mL} = 70 imes 10^{-6} ext{ m鲁} \]

02

Calculate Mass of Benzene

Using the density (\( \rho = 880 \text{ kg/m}^3 \)), we calculate the mass of 70 mL of benzene. The formula is:\[ m = \rho \times V = 880 \text{ kg/m}^3 \times 70 \times 10^{-6} \text{ m}^3 \]Calculate the mass:\[ m = 0.0616 \text{ kg} \]

03

Avogadro's Number and Molar Mass

Find the mass of one benzene molecule using its molar mass (\( M = 78 \text{ kg/kmol} \)) and Avogadro's number (\( N_A = 6.02 \times 10^{26} \text{ kmol}^{-1} \)):\[ m_0 = \frac{M}{N_A} = \frac{78 \text{ kg/kmol}}{6.02 \times 10^{26} \text{ kmol}^{-1}} \]Calculate the mass of one molecule:\[ m_0 = 1.30 \times 10^{-25} \text{ kg} \]

04

Calculate Number of Molecules

The number of benzene molecules is found by dividing the total mass by the mass of a single benzene molecule:\[ \, \text{Number of molecules} = \frac{m}{m_0} = \frac{0.0616 \text{ kg}}{1.30 \times 10^{-25} \text{ kg}} \]Calculate the number of molecules:\[ \, \text{Number of molecules} = 4.8 \times 10^{23} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density Conversion

When dealing with different measurements, it's often necessary to convert between units to use given formulas correctly. This is especially true in chemistry and physics. Consider density. In this example, you're presented with density in \( \text{kg/m}^3 \) but given a volume in milliliters (mL). To find mass accurately, you first need to convert the volume from mL to cubic meters (m鲁). This direct conversion is necessary since 1 mL equals 1 cm鲁, and further, 1 cm鲁 is equal to 1脳10鈦烩伓 m鲁. So, for 70 mL of benzene, the volume becomes 70 脳 10鈦烩伓 m鲁, making calculations coherent with the density units.

Mass Calculation

Once you have the converted volume, calculating the mass is straightforward using the formula for mass: \( m = \rho \times V\). Here, \( \rho \) denotes density, while \( V \) is volume. With benzene, the density is known as \( 880 \, \text{kg/m}^3\), and the converted volume is \( 70 \times 10^{-6} \, \text{m}^3\). Multiplying these values gives the mass of benzene:

• Mass = Density 脳 Volume
• \[ m = 880 \, \text{kg/m}^3 \times 70 \times 10^{-6} \, \text{m}^3 = 0.0616 \, \text{kg} \\]

Applying this formula helps understand the foundational process of translating physical quantities into understandable numbers, setting up the problem for further analysis.

Avogadro's Number

Avogadro's number is a crucial constant in chemistry, representing the number of constituent particles (usually atoms or molecules) in one mole of any substance. This number is roughly \(6.02 \times 10^{23}\). Here, Avogadro's number allows us to calculate the mass of a single benzene molecule.Subsequently, we use it together with the molar mass to find the individual molecule's mass, setting the stage for counting benzene molecules within the sample. This understanding makes Avogadro's number indispensable in bridging large scale reactions with molecular-scale events.

Molar Mass

Molar mass is the mass of one mole of a substance, usually presented in g/mol or kg/kmol. For benzene, the molar mass is given as \( 78 \, \text{kg/kmol} \). To find the mass of an individual molecule, we divide this molar mass by Avogadro's number. This calculation is crucial since it represents the transition from macro to micro scale. The formula is:

• \[ m_0 = \frac{M}{N_A} = \frac{78 \, \text{kg/kmol}}{6.02 \times 10^{26} \, \text{kmol}^{-1}} = 1.30 \times 10^{-25} \, \text{kg}\]

This step is essential in quantifying the number of individual molecules within any given sample, paving the way for accurate predictions in further computations.

Volume Conversion

Finally, revisiting the initial volume conversion in our calculations is critical. Before calculating mass, it's important to ensure that all measurements align. For chemistry calculations involving density, matching units in mass and volume ensures consistency and accuracy. Remember, 1 mL equals 1 cm鲁, and further conversion to m鲁 utilizes the ratio \(1 \, \text{cm}^3 = 1 \times 10^{-6} \, \text{m}^3\). So, when calculating through conversions, always check:

• Volume in compatible units with density.
• Whether the conversion aligns results with desired units like kg or m鲁.

Such careful conversion is necessary to confirm that subsequent calculations hold true rigorously.