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Suppose that the change in V\(_m\) was caused by the entry of Ca\(^{2+}\) instead of Na\(^+\). How many Ca\(^{2+}\) ions would have to enter the cell per unit membrane to produce the change? (a) Half as many as for Na\(^+\); (b) the same as for Na\(^+\); (c) twice as many as for Na\(^+\); (d) cannot say without knowing the inside and outside concentrations of Ca\(^{2+}\).

Short Answer

Expert verified
(a) Half as many as for Na\(^+\).

Step by step solution

01

Understand the question

We need to determine how the entry of Ca\(^{2+}\) ions affects the change in membrane potential \(V_m\) in comparison to Na\(^+\) ions. Different options are given based on how many Ca\(^{2+}\) ions are needed relative to Na\(^+\).
02

Recall the charge difference

Ca\(^{2+}\) ions have a charge of +2, compared to Na\(^+\) ions which have a charge of +1. This means each Ca\(^{2+}\) ion carries double the charge compared to each Na\(^+\) ion.
03

Determine required ions

Since Ca\(^{2+}\) is doubly charged, the entry of one Ca\(^{2+}\) ion would produce the same charge difference as two Na\(^+\) ions. Therefore, half as many Ca\(^{2+}\) ions are needed to effect the same change in \(V_m\) as Na\(^+\) ions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calcium Ions
Calcium ions, denoted as Ca\(^{2+}\), are positively charged ions that play a crucial role in various cellular processes. These ions carry a positive charge of +2, which means they are doubly charged compared to other common cations like sodium ions. This double charge is significant in the context of membrane potential, as it impacts how these ions influence charge distribution and changes in potential.

In cellular environments, calcium ions are often found in low concentrations inside cells but higher concentrations outside. This gradient is essential for signal transduction and muscle contraction. When calcium ions enter a cell, they can trigger various responses, such as the release of neurotransmitters or the initiation of muscle contraction.
Sodium Ions
Sodium ions, or Na\(^+\), are fundamental to maintaining the cell's resting membrane potential and facilitating action potentials. With a single positive charge, sodium ions are involved in transporting other ions or molecules across cell membranes.

In neurons, the rapid influx of sodium ions during an action potential leads to depolarization, a critical step in the transmission of nerve impulses. Sodium-potassium pumps work to maintain a high concentration of sodium ions outside cells and potassium ions inside, which is vital for various cellular functions.
  • Sodium ions have a +1 charge.
  • Key players in nerve impulse transmission.
  • Integral to maintaining balance in ion concentrations and membrane potential.
Ion Concentration
Ion concentration refers to the amount of specific ions present in and around a cell. This concentration plays a pivotal role in determining the membrane potential, which is essentially the voltage difference across a cell's membrane.

The membrane potential is influenced by the differential distribution of ions like calcium and sodium across the membrane, as quantified by the Nernst equation. This equation considers the intracellular and extracellular concentrations of ions to calculate the equilibrium potential.

Changes in ion concentration can lead to alterations in membrane potential, thereby impacting cellular activities such as muscle contraction, nerve firing, and the opening or closing of ion channels.
Electrochemical Gradient
The electrochemical gradient is a combination of two forces: the chemical gradient and the electrical gradient. It drives the movement of ions across membranes.

The chemical gradient refers to the difference in ion concentration between the inside and outside of the cell, whereas the electrical gradient is the difference in charge across the membrane.
  • Driving force for ion movement.
  • Maintains crucial cellular activities.
  • Essential for establishing membrane potential.

These gradients are at the heart of many physiological processes, including the transmission of nerve impulses and the operation of cardiac muscles. By maintaining a balance between these gradients, cells can ensure proper function and responsiveness to external stimuli.

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Most popular questions from this chapter

Electric eels and electric fish generate large potential differences that are used to stun enemies and prey. These potentials are produced by cells that each can generate 0.10 V. We can plausibly model such cells as charged capacitors. (a) How should these cells be connected (in series or in parallel) to produce a total potential of more than 0.10 V? (b) Using the connection in part (a), how many cells must be connected together to produce the 500-V surge of the electric eel?

In one type of computer keyboard, each key holds a small metal plate that serves as one plate of a parallel-plate, airfilled capacitor. When the key is depressed, the plate separation decreases and the capacitance increases. Electronic circuitry detects the change in capacitance and thus detects that the key has been pressed. In one particular keyboard, the area of each metal plate is 42.0 mm\(^2\), and the separation between the plates is 0.700 mm before the key is depressed. (a) Calculate the capacitance before the key is depressed. (b) If the circuitry can detect a change in capacitance of 0.250 pF, how far must the key be depressed before the circuitry detects its depression?

A budding electronics hobbyist wants to make a simple 1.0-nF capacitor for tuning her crystal radio, using two sheets of aluminum foil as plates, with a few sheets of paper between them as a dielectric. The paper has a dielectric constant of 3.0, and the thickness of one sheet of it is 0.20 mm. (a) If the sheets of paper measure 22 \(\times\)28 cm and she cuts the aluminum foil to the same dimensions, how many sheets of paper should she use between her plates to get the proper capacitance? (b) Suppose for convenience she wants to use a single sheet of posterboard, with the same dielectric constant but a thickness of 12.0 mm, instead of the paper. What area of aluminum foil will she need for her plates to get her 1.0 nF of capacitance? (c) Suppose she goes high-tech and finds a sheet of Teflon of the same thickness as the posterboard to use as a dielectric. Will she need a larger or smaller area of Teflon than of posterboard? Explain.

A parallel-plate air capacitor is to store charge of magnitude 240.0 pC on each plate when the potential difference between the plates is 42.0 V. (a) If the area of each plate is 6.80 cm\(^2\), what is the separation between the plates? (b) If the separation between the two plates is double the value calculated in part (a), what potential difference is required for the capacitor to store charge of magnitude 240.0 pC on each plate?

A parallel-plate air capacitor is made by using two plates 12 cm square, spaced 3.7 mm apart. It is connected to a 12-V battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field between the plates? (d) What is the energy stored in the capacitor? (e) If the battery is disconnected and then the plates are pulled apart to a separation of 7.4 mm, what are the answers to parts (a)-(d)?

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