A spherical balloon of radius 3 cm containing helium gas has a pressure of 48 x 10^-3 bar. At the sa

Q: Why do we use Boyle's Law here?

Because the problem states 'same amount of gas' and 'same temperature', meaning the number of moles ($n$) and temperature ($T$) are constant. Boyle's Law specifically deals with this scenario.

Q: Do I need to convert the radii from cm to meters?

No, because we are taking a ratio of the radii ($r_1/r_2$), the units will cancel out as long as they are the same.

Q: Why didn't we calculate the exact volume of the first balloon?

Calculating the exact volume introduces messy decimals and $\pi$. By keeping it as an expression, the constants cancel out, making the math much cleaner and less prone to errors.

Q: How did $0.75 \times 10^{-3}$ become $750 \times 10^{-6}$?

To change the exponent from $-3$ to $-6$, we are effectively dividing the multiplier by $10^3$ (or $1000$). To keep the value the same, we must multiply the base number by $1000$. So, $0.75 \times 1000 = 750$.

Q: What if the temperature had changed?

If the temperature changed, we would use the Combined Gas Law: $(p_1 V_1)/T_1 = (p_2 V_2)/T_2$.

Question

A spherical balloon of radius $3 	ext{ cm}$ containing helium gas has a pressure of $48 	imes 10^{-3} 	ext{ bar}$. At the same temperature, the pressure, of a spherical balloon of radius $12 	ext{ cm}$ containing the same amount of gas will be ......... $	imes 10^{-6} 	ext{ bar}$.

Sigma Percentile · Accepted Answer

### The Setup: Expanding the Balloon

Imagine you are holding a small spherical balloon with a radius of $3 	ext{ cm}$. Inside this balloon, helium gas is bouncing around, exerting a pressure of $48 	imes 10^{-3} 	ext{ bar}$ on the inner walls.

Now, imagine a magical process where we transfer all of this exact same helium gas into a much larger spherical balloon, one with a radius of $12 	ext{ cm}$. Crucially, we do this without changing the temperature of the gas. Our mission is to find the new pressure inside this larger balloon.

### The Master Equation: Boyle's Law

In the world of gases, whenever we see the phrases "same amount of gas" and "same temperature," an alarm bell should ring in our heads: **Boyle's Law**.

Boyle's Law states that for a fixed mass of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional. Mathematically, this is beautifully expressed as:

$$p_1 V_1 = p_2 V_2$$

This equation is our master key. It tells us that whatever the gas loses in pressure, it makes up for in volume, keeping their product perfectly constant.

### The Elegance of Cancellation

Before we rush to plug in numbers, let's think about the volumes. The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$. Let's substitute this raw expression into our master equation for both balloons:

$$p_1 \left( \frac{4}{3} \pi r_1^3 ight) = p_2 \left( \frac{4}{3} \pi r_2^3 ight)$$

Here is where the magic of algebra happens. Notice that the constant factor $\frac{4}{3}\pi$ appears on both sides of the equation. We can completely cancel it out! This is why we never calculate intermediate values if we don't have to. By keeping the expressions raw, we save ourselves from messy decimal calculations and potential errors.

$$p_1 r_1^3 = p_2 r_2^3$$

Now, we can easily isolate our unknown final pressure, $p_2$:

$$p_2 = p_1 \left( \frac{r_1}{r_2} ight)^3$$

### The Final Calculation

With our streamlined equation ready, we can finally substitute our given values. We know $p_1 = 48 	imes 10^{-3} 	ext{ bar}$, $r_1 = 3 	ext{ cm}$, and $r_2 = 12 	ext{ cm}$.

$$p_2 = 48 	imes 10^{-3} 	imes \left( \frac{3}{12} ight)^3$$

The ratio inside the parenthesis simplifies wonderfully:

$$p_2 = 48 	imes 10^{-3} 	imes \left( \frac{1}{4} ight)^3$$

Cubing $\frac{1}{4}$ gives us $\frac{1}{64}$.

$$p_2 = 48 	imes 10^{-3} 	imes \frac{1}{64}$$

$$p_2 = \frac{48}{64} 	imes 10^{-3}$$

Both $48$ and $64$ are divisible by $16$, reducing the fraction to $\frac{3}{4}$, which is $0.75$.

$$p_2 = 0.75 	imes 10^{-3} 	ext{ bar}$$

We are almost there! The question specifically asks for the answer in the format of $\dots 	imes 10^{-6} 	ext{ bar}$. To convert our answer, we multiply the decimal by $1000$ and compensate by decreasing the exponent by $3$:

$$p_2 = 750 	imes 10^{-6} 	ext{ bar}$$

Thus, our final integer answer is **750**.

Enter Numerical Value:

Visualized Solution

Visualizing the Setup

Boyle's Law: $p_{1} V_{1} = p_{2} V_{2}$

Substituting $V = \frac{4}{3} π r^{3}$

Canceling $\frac{4}{3} π$

Isolating $p_{2}$

Plugging in $r_{1}, r_{2}, p_{1}$

Simplifying the Ratio

Final Answer Formatting

The Way Forward

The Sigma Insight: Gaseous State

The Setup

The Master Equation

The Elegance of Cancellation

The Final Calculation

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