Chapter 1: Problem 72
A human brain weighs about \(1 \mathrm{~kg}\) and contains about \(10^{11}\) cells. Assuming that each cell is completely filled with water (density \(=1 \mathrm{~g} / \mathrm{mL}\) ), calculate the length of one side of such a cell if it were a cube. If the cells are spread out into a thin layer that is a single cell thick, what is the surface area in square meters?
Short Answer
Expert verified
Each cell's side is approximately 0.0215 cm, and the surface area is about 4622.5 m².
Step by step solution
01
Determine the total volume of the brain cells
Since the brain's density is about the same as water and the brain weighs approximately 1 kg, the total volume of the brain is 1 liter. As 1 liter is equal to 1000 mL and each milliliter of water has a mass of 1 gram, the total volume of the brain cells can also be considered as 1000 mL or 1000 cm³.
02
Calculate the volume of a single cell
There are approximately \(10^{11}\) cells in the brain. To find the volume of a single cell, divide the total volume by the number of cells:\[\text{Volume of a single cell} = \frac{1000 \text{ cm}^3}{10^{11}} = 10^{-8} \text{ cm}^3\]
03
Find the length of one side of a cubic cell
Assuming each cell is a cube, the volume of a cube is given by \(a^3\), where \(a\) is the length of a side. Set the volume of the cube equal to the cell's volume:\[a^3 = 10^{-8} \text{ cm}^3\]To find \(a\), take the cube root of both sides:\[a = (10^{-8})^{1/3} = 10^{-8/3} = 10^{-2.67} \approx 0.0215 \text{ cm}\]
04
Calculate the surface area of the spread-out cells
If the cells form a single cell thick layer, the total number of cells \(10^{11}\) gives the number of squares (each being 0.0215 cm on each side) on the surface. Thus, the total area \(A\) is:\[A = 10^{11} \times (0.0215 \text{ cm})^2\]First, calculate the area of one cell:\[(0.0215 \text{ cm})^2 = 0.00046225 \, \text{cm}^2\]Multiply by the total number of cells:\[A = 10^{11} \times 0.00046225 \, \text{cm}^2 = 4.6225 \times 10^7 \, \text{cm}^2\]Convert square centimeters to square meters \((1 \text{ m}^2 = 10,000 \text{ cm}^2)\):\[A = \frac{4.6225 \times 10^7 \, \text{cm}^2}{10^4} = 4622.5 \, \text{m}^2\]
Over 22 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Volume Calculation
Understanding volume is crucial in solving various Chemistry Problems. It's all about measuring how much space an object or substance takes up. In this exercise, we're calculating the volume of a human brain, considering it consists of many small compartments: the cells.
The brain weighs approximately 1 kilogram. Since its density is nearly the same as water, we know its total volume is about 1 liter. In scientific terms, this equates to 1000 milliliters or, equivalently, 1000 cubic centimeters ( cm^3 ). Calculating the volume allows chemists and scientists to break down intuitive concepts into specific numbers, which can then be further explored.
When we talk about the volume of each individual cell, it's about dividing the total brain volume by the number of individual cells present. In this scenario, the brain contains about 10 to the power of 11 cells, making each cell's volume 10 to the power of negative 8 cubic centimeters. This process emphasizes how even in large structures, individual parts like cells can be incredibly tiny.
The brain weighs approximately 1 kilogram. Since its density is nearly the same as water, we know its total volume is about 1 liter. In scientific terms, this equates to 1000 milliliters or, equivalently, 1000 cubic centimeters ( cm^3 ). Calculating the volume allows chemists and scientists to break down intuitive concepts into specific numbers, which can then be further explored.
When we talk about the volume of each individual cell, it's about dividing the total brain volume by the number of individual cells present. In this scenario, the brain contains about 10 to the power of 11 cells, making each cell's volume 10 to the power of negative 8 cubic centimeters. This process emphasizes how even in large structures, individual parts like cells can be incredibly tiny.
Density and Mass Relationship
The relationship between density and mass is a foundational concept in science. Density refers to how much mass is contained in a particular volume. In this problem, the brain's density is considered to be similar to water, making calculations straightforward.
A key detail is that the brain, at 1 kilogram, weighs the same as 1000 milliliters or 1 liter of water. This is because the density of water is 1 gram per milliliter ( 1 ext{ g/mL} ). By knowing the density, one can easily transition between mass and volume. Consider how, by knowing the mass (1 kg), we quickly deduce the volume (1000 mL), using the density as the linking factor.
When we think of each brain cell as being completely filled with water, we seamlessly use these mass and volume relationships to estimate the size and characteristics of a single cell. This illustrates how density serves as a bridge between understanding mass and size across a wide range of fields.
A key detail is that the brain, at 1 kilogram, weighs the same as 1000 milliliters or 1 liter of water. This is because the density of water is 1 gram per milliliter ( 1 ext{ g/mL} ). By knowing the density, one can easily transition between mass and volume. Consider how, by knowing the mass (1 kg), we quickly deduce the volume (1000 mL), using the density as the linking factor.
When we think of each brain cell as being completely filled with water, we seamlessly use these mass and volume relationships to estimate the size and characteristics of a single cell. This illustrates how density serves as a bridge between understanding mass and size across a wide range of fields.
Cell Structure and Measurement
Cells are fascinating and complex. In the context of this exercise, they are simplified into cubes for easier calculations. This assumption aids in teaching basic geometric and atomic principles.
The volume of a cell, previously calculated as 10^{-8} ext{ cm}^3 , allows us to deduce the length of one side of a cubic cell. By applying algebraic skills, particularly cube root calculations, we find the length to be around 0.0215 centimeters. This might sound tiny, but in the vast world of cells, it provides valuable insights into average cell dimensions.
Using such simplified models helps students get a grasp of spatial reasoning on a microscopic scale. By understanding the average measurements of a cell, it paves the way for more advanced biological and chemical discussions. Beyond just numbers, it illustrates a fundamental piece of biology's story – from understanding the basic building blocks to how they form more complex structures.
The volume of a cell, previously calculated as 10^{-8} ext{ cm}^3 , allows us to deduce the length of one side of a cubic cell. By applying algebraic skills, particularly cube root calculations, we find the length to be around 0.0215 centimeters. This might sound tiny, but in the vast world of cells, it provides valuable insights into average cell dimensions.
Using such simplified models helps students get a grasp of spatial reasoning on a microscopic scale. By understanding the average measurements of a cell, it paves the way for more advanced biological and chemical discussions. Beyond just numbers, it illustrates a fundamental piece of biology's story – from understanding the basic building blocks to how they form more complex structures.
Mathematical Conversions in Science
Mathematical conversions are central to navigating science problems efficiently. They help in switching units to better fit the context and deliver answers in universally accepted formats.
Consider the problem's transition from cubic centimeters to square meters for surface area calculations. The calculated spread-out cells form a layer totaling an area of 4.6225 million square centimeters. This conversion emphasizes grasping the larger picture by converting from smaller units to larger ones — in this case, to square meters ( 1 ext{ m}^2 = 10,000 ext{ cm}^2 ).
Mastery of conversions opens doors to comprehensive analysis and ensures that, whether in Chemistry Problems or various scientific fields, solutions are applicable and meaningful. Practical knowledge of these conversions is indispensable for students, empowering their analytical skills in both academic and real-world applications.
Consider the problem's transition from cubic centimeters to square meters for surface area calculations. The calculated spread-out cells form a layer totaling an area of 4.6225 million square centimeters. This conversion emphasizes grasping the larger picture by converting from smaller units to larger ones — in this case, to square meters ( 1 ext{ m}^2 = 10,000 ext{ cm}^2 ).
Mastery of conversions opens doors to comprehensive analysis and ensures that, whether in Chemistry Problems or various scientific fields, solutions are applicable and meaningful. Practical knowledge of these conversions is indispensable for students, empowering their analytical skills in both academic and real-world applications.
