The pressure needed to make synthetic diamonds from graphite is \(8 \times 10^{4}\) atm. Express this pressure in (a) Pa; (b) kbar; (c) Torr; (d) lb \(^{\text {inch }}{ }^{-2}\).

Understanding how to convert atmospheric pressure to Pascals is fundamental for many scientific calculations. The standard atmosphere (atm) is a unit of pressure defined as being precisely equivalent to 101,325 Pascals (Pa). This exactitude means that converting from atmospheres to Pascals is straightforward: simply multiply the value in atmospheres by 101,325.

To exemplify, converting a pressure of 8 atmospheres, which might be required to synthesize diamonds, entails multiplying 8 by 101,325. The resulting figure is the pressure in Pascals: \[8 \times 10^{4} \text{atm} \times 101,325 \text{Pa/atm}\].

This mechanical process exemplifies the direct proportionality between atmospheres and Pascals. For practical problem-solving, ensure to maintain precision by accounting for scientific notation during conversion.

Pressure conversion from atmospheres to kilobars is another common task in the realm of physical sciences. Unlike Pascals, kilobars (kbar) are a multiple of the bar, which is another standard unit of pressure. Here, 1 bar equals 100,000 Pascals, and consequently, 1 kilobar equals 1,000 bars. Since 1 atmosphere is approximately 1.01325 bars, converting atm to kbar involves first multiplying by 1.01325 to get bars, and then dividing the result by 1,000.

For instance, a pressure of 8 atmospheres is equal to \[8 \times 10^{4} \text{atm} \times 1.01325 \text{bar/atm} / 1,000\] kilobars. By using this method, scientific calculations which require units of kilobars become accessible and precise.

Converting from atmospheres to Torr is also a routine conversion in pressure-related studies. Torricelli's unit, Torr, is named after the Italian physicist who developed the barometer. It is a manometric unit that equates to one millimeter of mercury (mmHg). One atm approximately equals 760 Torr, making this conversion a matter of simple multiplication.

Using our example, a pressure needed in the synthesis of diamonds, which is 8 atmospheres, converts to Torr by multiplying: \[8 \times 10^{4} \text{atm} \times 760 \text{Torr/atm}\].

This conversion is vital in situations where precise pressure measurements are required, such as in the synthesis of materials and in various experimental setups.

The transition from atmospheres to pounds per square inch (psi) is pivotal for many engineering applications, where pressure needs to be understood in terms of its effect on surfaces. The psi is the pressure resultant from one pound-force applied to an area of one square inch. Since one atm is equal to 14.696 psi, converting atm to psi involves multiplying the atm value by 14.696.

For example, the pressure to create synthetic diamonds which is 8 atmospheres is equivalent to \[8 \times 10^{4} \text{atm} \times 14.696 \text{lb/in}^{2}/\text{atm}\] pounds per square inch. Engineers and scientists utilize this conversion for practical applications, such as hydraulic system design, tire pressure maintenance, and even in scuba diving equipment.