A cube of gold that is 1.00 cm on a side has a mass of 19.3 g. A single gold atom has a mass of 197.0 u. (b) From the information given, estimate the diameter in Å of a single gold atom.

검증된 단계별 안내

Step 1: First, we need to calculate the volume of the gold cube. The volume of a cube is given by the formula \(V = a^3\), where \(a\) is the length of one side of the cube. In this case, \(a = 1.00\) cm.

Step 2: Next, we need to calculate the number of gold atoms in the cube. We know that the mass of the cube is 19.3 g and the mass of a single gold atom is 197.0 u. We can convert the mass of the cube to atomic mass units (u) by using the conversion factor 1 g = 6.022 x 10^23 u (Avogadro's number). Then, we divide the total mass of the cube by the mass of a single atom to find the number of atoms.

Step 3: Now, we can calculate the volume occupied by a single gold atom. We divide the total volume of the cube by the number of atoms to get the volume of a single atom.

Step 4: We know that atoms are roughly spherical, so we can use the formula for the volume of a sphere \(V = \frac{4}{3}πr^3\) to find the radius of a gold atom. We can solve this equation for \(r\) to get \(r = \left(\frac{3V}{4π}\right)^{\frac{1}{3}}\).

Step 5: Finally, we can find the diameter of a gold atom by multiplying the radius by 2. Remember to convert the diameter from cm to Å by using the conversion factor 1 cm = 10^8 Å.

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주요 개념

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Density

Density is defined as mass per unit volume and is a crucial property of materials. In this case, the density of gold can be calculated using the mass of the cube and its volume. Understanding density allows us to relate the macroscopic properties of a substance to its atomic structure, which is essential for estimating atomic dimensions.

Avogadro's Number

Avogadro's number, approximately 6.022 x 10²³, is the number of atoms or molecules in one mole of a substance. This concept is vital for converting between the macroscopic scale (grams) and the atomic scale (individual atoms). By using Avogadro's number, we can determine how many gold atoms are present in the given mass, which aids in estimating the size of a single atom.

Atomic Radius

The atomic radius is a measure of the size of an atom, typically defined as the distance from the nucleus to the outermost electron shell. In this context, estimating the diameter of a gold atom involves understanding its atomic radius and how it relates to the number of atoms in a given volume. This concept is fundamental for translating the mass and volume of gold into the dimensions of individual atoms.