Problem 8 - University Physics with Modern Physics

Question

Black smokers are hot volcanic vents that emit smoke deep in the ocean floor. Many of them teem with exotic creatures, and some biologists think that life on earth may have begun around such vents. The vents range in depth from about 1500 m to 3200 m below the surface. What is the gauge pressure at a 3200-m deep vent, assuming that the density of water does not vary? Express your answer in pascals and atmospheres.

Step-by-Step Solution

Answer

The gauge pressure is approximately 31,392,000 Pa or 309.71 atm.

Step 1: Understanding the Problem

We need to calculate the gauge pressure at a depth of 3200 meters in the ocean. Gauge pressure is the pressure relative to atmospheric pressure and is often calculated for submerged objects using the formula \( P = \rho g h \), where \( \rho \) is the density of water, \( g \) is the acceleration due to gravity, and \( h \) is the depth.

Step 2: Parameters and Constants

The density of water \( \rho \) is typically about \( 1000 \, \text{kg/m}^3 \). The acceleration due to gravity \( g \) is approximately \( 9.81 \, \text{m/s}^2 \). The depth \( h \) is given as 3200 meters.

Step 3: Calculate Gauge Pressure in Pascals

Using the formula \( P = \rho g h \), substitute the values: \( \rho = 1000 \, \text{kg/m}^3 \), \( g = 9.81 \, \text{m/s}^2 \), and \( h = 3200 \, \text{m} \).\[ P = 1000 \times 9.81 \times 3200 = 31,392,000 \, \text{Pa} \] This is the gauge pressure in pascals.

Step 4: Convert Gauge Pressure to Atmospheres

1 atmosphere is equivalent to approximately \( 101,325 \, \text{Pa} \). Convert the calculated pressure from pascals to atmospheres:\[ P_{atm} = \frac{31,392,000}{101,325} \approx 309.71 \, \text{atm} \] This is the gauge pressure in atmospheres.

Key Concepts

Density of Water

The density of water is a fundamental property that plays a crucial role in various calculations involving fluids, especially in physics and engineering. Here, it's essential for determining the pressure exerted by water at different depths in the ocean.

• The standard value for the density of water is typically \(1000 \, \text{kg/m}^3\), which allows us to perform calculations with quite a bit of precision.
• This value assumes pure water at a temperature around 4°C, where water is densest. Real-world conditions like salinity and temperature can affect density, but for basic problem-solving, \(1000 \, \text{kg/m}^3\) is a convenient approximation.
• Understanding water density is crucial because it impacts buoyancy, pressure under water, and even how sound travels through water.

For calculations like those involving gauge pressure at ocean depths, this density helps calculate the weight of a column of water above a point. This weight directly impacts the pressure exerted at that depth.

Acceleration Due to Gravity

The acceleration due to gravity, denoted as \(g\), is the rate at which objects accelerate towards Earth. It's a critical component in calculating various forces and pressures, including those underwater.

• The commonly accepted value of \(g\) is \(9.81 \, \text{m/s}^2\), representing the acceleration of an object in freefall near the Earth's surface.
• This value can slightly vary depending on your location on Earth because the planet is not a perfect sphere and differences in altitude can affect gravity.
• In underwater pressure calculations, such as for gauge pressure, \(g\) helps determine the force exerted by the water's weight.

Incorporating \(g\) into the formula for pressure \(P = \rho g h\) means we account for how quickly objects, including water, are pulled towards the Earth, influencing how pressure increases with depth.

Pascal to Atmosphere Conversion

In many scientific and engineering problems, it's essential to convert between different units of pressure to understand or communicate results effectively. One common conversion is from pascals to atmospheres.

• A pascal (Pa) is the SI unit of pressure, defined as one newton per square meter. It's suitable for large-pressure calculations but is often more informative to express pressures in atmospheres (atm) for everyday understanding.
• One atmosphere is defined as \(101,325 \, \text{Pa}\), which approximates the average atmospheric pressure at sea level on Earth.
• To convert from pascals to atmospheres, you divide the pressure value in pascals by \(101,325\).

This conversion is evident in the solution of the gauge pressure at a deep-sea vent, where an understanding of both units helps clarify the extreme conditions deep underwater compared to surface pressures.